Given that there are multiple mutually exclusive conceptions of morality, both religious ones and non-religious ones, and the fact that in times of exponential growth of science atheistic forms of moralities can definitely be perceived as science-friendly religions as they usually do not contain as many dogmas as religions do. But fundamentally different conceptions of morality are no different from religions except for the fact that they do not assume existence of supernatural beings.
Most importantly we are lacking any scientific proof for any single one of these conception of moralities as well as the the existence of universal morality.
Especially, all these atheistic conceptions of morality are based on beliefs that are science-friendly.
In other words, religion is just a pejorative term for science-unfriendly beliefs whereas moralities are kinds of beliefs that are science-friendly.
So, asking about the scientific proof for any particular conception of morality is the same calibre of question as asking for the proof of the existence of God(s).
Science does not offer solutions for these kind of question, so why do we assume that any conception of morality even exist if we have no proof for it?
Because it serves our needs?
Because we act as if any morality actually existed?
Because it helps to create moral norms upon which legal norms could be created that in turn are needed for the society to exist?
On purely scientific grounds, one would think that these moralities are just a bunch of beliefs that are merely religions without God(s). Just that. All conceptions of moralities are as delusional as religions. The difference between moralities and religion are in detail. They are fundamentally the same as they bring a conception of Good and Evil. Just that.
I think the first really good definition of religion was from Durkheim, founder of academic sociology:
“A religion is a unified system of beliefs and practices relative to
sacred things, that is to say, things set apart and forbidden – beliefs
and practices which unite in one single moral community called a
Church, all those who adhere to them”
It takes understanding the wide variation of superstitious & magical practices, & how different say Wuism or Sami shamanism are from Abrahamic practices, to appreciate why this is a good definition. Durkheim is drawing attention to religion's social role, rather than epistemology or cosmology. Say in festivals or prayer or offerings, we see enactments of values, through manifesting shared attitudes towards what is sacred, & that is the source of the social glue of religion, in this picture.
Crucially, things like habeus corpus, scientific method, free speech rights, or the universal declaration of human rights meet this criteria. But a system of thought like say utilitarianism, or effective altruism, do not. With the former set, you have a kind of 'social immune response' build up, to violations of what a community holds sacred, & failure to maintain the sacred value risks dissolution of the community bound by that community holding it sacred - a polity, the international scientific community, democracies, the UN, etc.
"Most importantly we are lacking any scientific proof for any single
one of these conception of moralities"
This has a set of problems. Morality is a branch of culture, seeking to 'prove' a particular culture superior, does not have a good history. There is the assumption there the purpose of morality is unambiguous, & people do say 'wellbeing' or 'net happiness', but when you dig in to what someone means by those things, you find a whole worldview is smuggled into a given persons definition. Sam Harris clumsily ignored the history of philosophy & chauvinism in his attempt at sciencifying morality, discussed here: Is Sam Harris's view of morality innovating? What philosophers innovated specifics on morality?
Jonathan Haidt with his Moral Foundations Theory, finds broad differences between mainly agrarian & mainly pastoral communities, & that disease prevalence & living near disputed borders during key development years tends to lead communities to emphasise the 'sanctity/purity' moral axis which is more associated with pastoral cultures & rightwing conservative politics. Your moral foundations will be shaped by your experiences while your brain is developing, up until about age 25, then tolerance of ambiguity & other critical factors tend to become fixed, eg. amplifying geographical partisan sorting between rural & urban.
"religion is just a pejorative term for science-unfriendly beliefs"
The religious expressions of many people, groups & areas has been entirely compatible with science. For instance Buddhism literally aims at a science of subjectivity & the mind focused on ending suffering, with the current Dalai Lama XIV saying:
“If scientific analysis were conclusively to demonstrate certain
claims in Buddhism to be false, then we must accept the findings of
science and abandon those claims.”
You chose a pejorative way to define religion, with no evidence you present anything essential to what religions are or how they work.
All conceptions of moralities are as delusional as religions.
Crucially moralities help us collaborate. As discussed by declaring a value sacred, & enacting the telling & preservation of that, a cultural reaction is generated against violating it. Look at human germline research in Korea & China, & the reaction against that - fulfilling international norms around human suffering & safety & ethics approval means continuing to be able to attend foreign conferences, pursue academic collaborations & many other aspects of being in the scientific community. There are myriad examples. Conform to the community norms, access the benefits of the community, violate & get excommunicated.
I look again to Durkheim for the difference between morality & religions:
"Religion is in a word the system of symbols by means of which society
becomes conscious of itself"
But religions seek to harness & shape the way in which we collaborate & bond together, specifically through sharing what we hold sacred. By an experimental process of what makes communities cohere, or not, usually led by 'religious entrepreneurs', or cultural ones. The more coherence, the more 'social immune system' against loss or change, though also potentially more fragility & risk of sudden collapse. This is a process of taking and acting on emergent culture, through rituals, beliefs, experiences, etc etc. The Golden Rule is a near cultural universal in traditions around the world, exactly because prompting people to enact it is so good for collaboration.
This answer responds to the following portion of your post:
So, asking about the scientific proof for any particular conception of morality is the same calibre of question as asking for the proof of the existence of God(s). Science does not offer solutions for these kind of questions, so why do we assume that any conception of morality even exists if we have no proof for it?
Do you assume that any (overarching) conception of morality exists?
Some people - religious and non-religious - might possess a clear conception of a moral code, but many do not.
People who have an interest in the issues of morality are probably more likely to acknowledge that there is no proof of any kind of 'objective' morality. An objective morality would exist independently of a mind, but given that morality is a mental construct (either human or divine), this can never be the case. Even a god's morality (if the morality is determined by the god and not adopted from some non-cognitive entity) would be subjective.
Any rigorously considered moral code or attitude to which a person adheres will likely be informed at some level by knowledge obtained thanks to science, but it is typically directed by a specific goal or goals.
For example, a person might decide that a fundamental, guiding goal of their morality should be the maximisation of wellbeing for as many people/animals as possible. The choices they make will in turn reflect this goal. They might adopt veganism for instance, and choose - wherever possible - to support products that have been produced by workers who are afforded adequate working conditions.
Another person might decide that their morality is primarily self-oriented, that they will act primarily in their own interests at all times. They enjoy meat, so they decide to eat it. They want to maximise the value of every dollar they earn, so they buy the cheapest products available.
Of course, not everyone sits down and examines their morality in such a fashion. Goals are often revealed by, rather than guiding of, behaviour.
Reflect upon your own actions. What do they say about your goals? What, upon examination, is "your" morality?
Answering this question requires clarifying a number of assumptions embedded in the question.
First, treating value systems as a "religion" is to invert the relationship that philosophy usually assigns to philosophy and religion. IE religious worldviews and value systems are a SUBSET of philosophical worldviews and value systems, rather than philosophies being versions of religion. The question captures the innate similarities between the two, but uses a different label than is conventional for their similarities.
Second, and more important, the question presumes a very naïve version of Scientism -- IE that only those things which are "proven" by science are real, or knowable. This presumption has several problems with it. Listing them:
2a) Nothing in science is ever "proven". Things get supported, increasing one's level of certainty about them, but "proven" is is a threshold that no empirical issue ever achieves.
2b) Science primarily is an investigative process, and much of what is being investigated is in the realm of speculative -- the question presumes that science operates off a "book of facts", when for most actual science, it is instead a collection of possibly suspect observations, and speculative hypotheses.
2c) Science cannot justify itself, or its methods, that is done philosophically in the field of epistemology. And multiple valid subject areas (history, aesthetics, mathematics, literary criticism) are legitimate fields of knowledge. Scientism's claim to the exclusivity of science as a method to obtain knowledge is both logically contradictory, and refuted by observation.
The question invalidly presupposes that only science provides knowledge (2c error), notes correctly that morality is at best a difficult subject to investigate scientifically (see my answer to this PSE question for the conditions under which one can develop a science of morality Is it possible to scientifically determine good and evil?), and then infers from these two suspect premises that morality therefore cannot be known.
The actual case is that morality is in the field of philosophy. Philosophy is those subjects which we do not know well enough to TURN into sciences or other specialty fields. The actual subjects of study of philosophic sub-fields like morality/ethics is generally in dispute, as are the methods of investigation, and the outcomes one wants to achieve. We have a variety of methods that people have used to derive moral and ethical theories, INCLUDING the revealed and intuitive methods of religion.
The advocates of different methods of doing morality have not yet been able to agree on much, but that does not mean that morality is meaningless, nonexistant, delusional, or any of the other strong negative inferences that the questioner postulates. Subjects that we do not yet fully understand are not therefore nonexistent subjects and fields.