MCMC Mixing Mysteries

I love doing Math Reviews, for the random stuff I get to read. Here is a paper I probably would not have found:
MR3208124 Rom\’an, Jorge Carlos, Hobert, James P.\ and Presnell, Brett, On reparametrization and the Gibbs sampler, Statist. Probab. Lett. 91 (2014), 110–116.

The Gibbs sampler may be out of fashion with the Bayesian computation crowd these days, but the reparameterizations are still mysterious. I tried the PyMC3 NUTS sampler on the first example and it too has rather different mixing times:

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