It seems that a proof of the conjecture "away from p = 2" has already been given: https://annals.math.princeton.edu/2023/197-1/p05

The same authors (and two other authors) have given a proof over Z here: https://arxiv.org/abs/2310.16399

I am not capable of evaluating whether the proofs of the latter paper are correct. I still take the article as evidence for YES, as based on other information it seems likely that they are.