Is the existence of qualia generally considered an unresolved problem by philosophers? Is there a consensus on its nature or whether it can be studied at all?
Purely philosophically, consciousness is stuck between two schools of thought (generally, since each idea also have different sub-theories).
See more on the wikipedia article on consciousness, especially the mind-body problem.
One is the materialist/realist idea, where consciousness is a product of cause and effect by random events dating back to the beginning of the universe with the big bang.
The other is the theist/deist/idealist idea, where consciousness is something beyond the physical, and has the power to influence our brain. Each school of thought presupposes something beyond the idea itself. Materialism presupposes that there is just matter and energy across time, and nothing beyond that. The converse presupposes that there exists some metaphysical higher plane outside of the purely physical, and that our mind/consciousness is either completely or partly residing within this higher plane, interacting with the physical plane.
The train of experiments that had it's outspringing in the original double-slit experiment seems to suggest that the mind is not only responsible for influencing the brain, but that it is quite literally responsible for manifesting it and the world around us. The mind might actually be more real than matter. In a way. InspiringPhilosophy on youtube has a video about this, where he goes through and explains the studies, with clips from interviews with experts.
I am not sure if the issue is resolved or not though. I think we are getting close.
First, let me clarify... Problems that are 'resolved' disappear from philosophy, in the sense that they are no longer discussed or analyzed. Philosophy is an analytical process that aims to restructure the way we think about particular topics or issues; it resolves when we have come to some consensus about what structure that topic or issue should have. That 'disappearing' might take different forms...
- A Wittgensteinian therapy, where we realize that we have made a mistake in language and stop asking the question
- The establishment of a paradigm, where the question is de facto resolved, and collapses into dogma, practice, and/or technique
- A dialectical synthesis in which a philosophical question transforms into a different (deeper) philosophical question
... but the upshot is that the philosophical question itself stops being a pressing or relevant concern. It may return or it may not: some questions ebb and flow like the tides over the course of generations, others merely evaporate. But the point is that we only philosophize about things that are active problematics.
Qualia and subjectivity are currently an active problematic. There have been several efforts over the last century to make the problematic 'disappear,' usually by making 'subjectivity' an invalid construct in on sense or another (E.g., Skinnerism and Logical Positivism). But the issue kept arising, because it lies at the heart of both empiricist and rationalist philosophy. There are ongoing efforts to try to bridge the brain/mind barrier in various branches of psychology and neurology, and they've had some successes, but they are a long, long way from rendering this problem 'resolved' in any meaningful philosophical sense.
As to whether or not the philosophers believe the issue can be resolved... Well, philosophers clearly study it, so they clearly think — or at least hope — that some resolution is possible. Few people waste their time on causes they believe to be entirely hopeless. What that resolution might look like is anyone's guess; or rather, that is precisely the nature of the debate, and we cannot know the answer until the debate runs its course. But clearly (again) they see value in the debate itself, and keep trying different approaches to analyzing and studying the problem. That part of your question more or less answers itself.
See https://qualiaresearchinstitute.org. It has very interesting things about this topic. But from just researching, it is trying to be resolved, but is still unresolved. Otherwise, I think science will help more with question than just pure philosophy.
I'm not sure which problem you're referring to, but if you're just asking whether there's general consensus on the nature or existence of consciousness, the answer is 'no'.
Here's an article from the Stanford Encyclopedia of Philosophy about the debates and academic research on consciousness up to the time the article was published. And here is another one on qualia. These are summaries of the state of the academic conversation up to 2014 and 2017 respectively. So, as of 2014 and 2017 respectively, no—no consensus. If you have more specific questions about more specific problems involving consciousness or qualia, the articles are helpfully divided into subsections about those problems.
The question is circular as you would have to be conscious of consciousness. At best you could define and redefine it, but in doing so you are left in a continuum of a point of view.
At best consciousness is intrinsically empty, assumes patterns and reuses those patterns as means to keep reassuming.
The reason I say consciousness is intrinsically empty is that the term means such a variety of things to so many schools.
Lucretius, in "The Nature of Things" equivocated (in terms of antiquity) consciousness as atomic facts reorganized again and again into patterns. Some reading into Wittgenstien could help, a little, but we are left with interpretations of interpretations in considering the question of consciousness.
At best consciousness is connecting and separating variables which as assumed are inverted into new variables at which the process is repeated. This ties in loosely Nietzches eternal reoccurrence with his perspectivism.
The problem is both resolved as defined and unresolved as open to further defintions.
Consciousness is not a problem in philosophy. It is only a problem in university philosophy, where it is not studied. The study of consciousness is called Mysticism or Yoga. Academics just talk about it and argue about their speculative theories.
Nobody in the perennial tradition talks about the 'problem of consciousness'. Rather, they talk about the problem of ignorance.
An answer to your question would require giving a bit more detail since the problem has many aspects and may be formulated in various ways. Chalmers' 'hard' problem, for instance, is easy to solve, but other related problems are more tricky.