It's an interesting problem, and you're not doing it justice.

A and B have a prior based on certain evidence. Their first guess conveys only the mean of that prior. You also posit that they have a shared belief about the (expected) amount of evidence behind their prior.

To update at each iteration, they need to infer what evidence about the world is behind the exchange of guesses so far.

I don't agree with anything you've claimed about this scenario. I'll grant you any simplifying assumptions you need to prove it, but let's be clear about what those assumptions are.