@jonah knows about that
Did you try different random seeds for four chains? I think you need initialize chains more carefully, or do some scaling of the parameters, Now some random initial values are way beyond anything sensible.
Yes! Also Pareto k’s are much better, and elpd_loo is much better, so all diagnostics tell negbin is much better than Poisson. I recommend also plotting the marginal posterior of phi.
I have previously used inverse-gamma prior, and you could try that, too, but to give a well justified recommendation would require some thinking or googling. I’ll get back to this in a few days unless you are able to find something before that.