"The ubiquity of this issue is evidenced by the paper introducing the Adam optimizer [35]. In the course of introducing an optimizer with strong empirical performance, it also offers a theorem regarding convergence in the convex case, which is perhaps unnecessary in an applied paper focusing on non-convex optimization. The proof was later shown to be incorrect in [63]."

I strongly disagree here. Researchers, please keep trying to prove that your proposed optimizers are correct, even if you can only manage to prove it for a subclass of problems, and even if it risks publishing an accidentally incorrect proof.