And this paper suggest even <1.01 rule! All these suggestions are ad-hoc starting points. Rhat and ESS (previously known as n_eff) are useful as they are scale free, that is, when checking them for many parameters you donâ€™t need to compare them to the standard deviation of the marginal posterior of that parameter or to the domain knowledge.

If you donâ€™t like arbitrariness of suggested Rhat and ESS thresholds, you can always in the end look at the Monte Carlo standard error (MCSE) for the quantities of interest and use the domain knowledge to assess whether the accuracy is sufficient. Stan and ArviZ computations use Rhat and ESS to compute MCSE, so you will get the benefit of the multi-chain diagnostic in MCSE, too, although MCSE estimates might be slightly overoptimistic (say half too small) if ESS (n_eff) is small.

Great!