Convergence of parameter transformations

Quick question: When monitoring convergence with e.g. the new rhat/ESS, do I need to worry if transformations of variables converge? For example, if I model a binomial outcome the parameter for log odds ratio converges quickly but odds ratio (e.g. exp(lor)) does not should I keep sampling until rhats are all good for the ORs as well?

Do you actually observe invertible univariate transforms of parameters not converging?

About multivariate functionals (e.g. the log-target density), here is a discussion that might be of interest.

I am going to have to look at this closer. I think I may have been mistaking indices. The linked discussion is exactly what I needed, marked as answered!