The answer to the question in your last paragraph is, Yes. Well, you could try falsifying the contradictory of a conjecture. But I assume that you are assuming even that wouldn't work.
There is a philosophical argument that all claims in the empirical sciences must be falsifiable. The catch is that a falsification of any scientific theory is always possible. In that context, it can be tempting to dismiss all scientific theories as "empty conjecture". But even in that context, a theory that is the best we have for now, and has not yet been falsified, is not an empty conjecture. You'll find an account of falsifiability theory at Falsifiabiliy - Wikipedia
But this issue is not really relevant to much of our everyday knowledge. You can verify or falsify the claim that it is raining by going outside, looking around and perhaps holding out your hand. There's nothing wrong with that - unless you have decided to belief in radical scepticism. Even so, you have the right to decide for yourself whether common sense or the radical sceptic is the more plausible theory, and common sense does at least allow the possibility of falsifying everyday statements.
You are assuming that all proof must start from axioms. That's a particular philosophical theory. There are alternatives. See Foundationalism - Stanford Encyclopedia of Philosophy
But in most arguments, you will usually start from agreed premises, which makes life less complicated. If you can't agree some premises to start from, you are talking to someone who doesn't really want to argue, so it would be best to save your breath. One does meet such people from time to time.
In any case, you shouldn't assume from the fact that you might find a flat earther whom you cannot persuade of their error in two sentences, that the idea that the earth is flat cannot be falsified. In fact, it can, in a number of ways. But it would take longer than five minutes and would probably involve more than sitting in a room and talking about it.
There are other subjects where falsifiability is not relevant. Mathematical statements are not falsifiable. But that doesn't mean they are empty conjecture. They are proved or disproved by a different process. The same is true of logically true statements in general. These are known as necessary, or a priori or analytic statements. Both these disciplines use axioms to start from, but that doesn't mean they are empty conjecture.
Again, most statements about values are unfalsifiable. That's because they cannot be evaluated in the same way as empirical statements. That does not mean that they are "empty conjectures". You yourself seem to think that being an empty conjecture is a bad thing, and I cannot falsify that claim.
The traditional idea of philosophy included all efforts to discern the truth about the world. But gradually as the enterprises we now call the sciences developed, they began to be called by that name; this happened in the 19th century CE. What is now called philosophy is what was left, which is, it is true, is something of a melting-pot of rather chaotic ideas. But it is a forum in which new disciplines can develop and grow into their own identities. Psychology is a recent example. You'll find an account of this process at History of science - Wikipedia