Just to mention that I think the settings where I’ve seen this happen with “real models” involve initializations far from the typical set where gradients are really weak, and so (with nontrivial probability) chains manage to wander down to absurdly low standard deviations during the initial exploration… and (sinister voice) once they go in, they never come out.
I’ve got a model on simulated data where this was happening as frequently as approximately 1 in 4 chains. Even when chains worked, during the first few iterations this model threw a surprising number of warnings about “multiplier is zero but must be positive” and “current metropolis proposal is about to be rejected” (sorry I don’t have the actual warning text in front of me). Switching to the “old” noncentered parameterization squashed all these warnings as well as the sticky-boundary issue.