It would seem that you believe that. So what is your proof?

Proof is only meaningful in a system of shared assumptions (physics) or axioms (logic).

The statement that "it's wrong to believe without proof", equivalently, that "a single correct set of beliefs is mandated given by your proof (=evidence) and assumptions (=prior)", is a logical consequence of the rules of Bayesian deduction.

If moral realists, or anyone else, doesn't agree to Bayesianism or another commonly understood framework of proof, logic, and common assumptions, then I'm not interested in talking to them (about moral realism).

From the more physicsy side [...]

I believe all those things (with very high probability, to be pedantic). I know Godel's incompleteness theorems.

Saying that some true things are formally unprovable in a logic system is not significant evidence that any specific unproven statement (eg moral realism) is in fact true just because it's not disproven. And the theorem doesn't apply to probabilistic belief systems modeling the physical universe: I can have an arbitrary degree of confidence in a belief, as long as it's not probability 1, without requiring logical proof.

However, the situation for moral realism isn't "unproven conjecture", it's more like "unformalized conjecture whose proponents refuse to specify what it actually means". At least that's the state of debate in this thread.

As to my explaining to you the details of why anyone should be a moral realist, I"m not interested in attempting that. I'm not a committed moral realist myself, and I'd just have to do a lot of reading myself to find a good description of what ask for. Sorry.

Yet you assign sufficient probability to moral realism to think it's worth discussing or reading about. Otherwise you'd have said from the start, "I agree with you that moral realism has no evidence for it, let's just drop the subject". To have such a high prior requires evidence. If you don't have such evidence, you are wrong.