> I know they are so close to a computationally-assisted proof of counterexample that it is virtually inevitable at this point.

That's a strong claim. Is it based on more than the linked work on some model problems from fluid mechanics?

I will say that I dread the discourse if it works out, since I don't believe enough people will understand that using a PINN to get new solutions of differential equations has substantially no similarity to asking ChatGPT (or AlphaProof etc) for a proof of a conjecture. And there'll be a lot of people trying to hide the difference.