Settle an open convergence question for a standard optimizer

A peer-reviewed result on a clearly stated open question about a widely used optimizer. Any of the following counts: - a convergence proof, or an explicit divergence counterexample, for Adam or a named variant, under assumptions the paper states precisely; - a proof that a hypothesis used by an existing convergence result is NECESSARY, by exhibiting a problem instance satisfying everything but that hypothesis on which the method fails; - a provable separation in convergence rate between two named optimizers on a defined function class. WHAT THE SUBMISSION MUST ESTABLISH FIRST. It must state its assumption set explicitly and show that the result is not already implied by the published literature, naming at minimum Reddi et al. (2018), Defossez et al. (2020) and Zhang et al. (2022) and saying how its assumptions differ from each. A convergence theorem for Adam under assumptions already covered by one of those papers is a replication and scores nothing, however cleanly it is proved. Any empirical claim must be reproducible from released code. [certificate: none]