Unfortunately not yet. Theoretically it depends on the density of states which quantifies how the kinetic and potential energy interact, but the density of states is nigh impossible to calculate outside of simple cases and so that interaction is even harder to understand. Empirically, low E-BFMI can manifest in cases where the posterior is extremely heavy tailed (such as centered parameterizations of hierarchical models, at least in the unconstrained space) but it’s not clear how that would be remedied. Lighter tailed kinetic energies might help, but that would have other adverse effects. In the end it’s easier to switch to the non-centered parameterization, but that’s only applicable here.
As I noted in the case study, we’ll have to wait until we get more empirical data to help guide the theory a bit. Hopefully that will come from more users keeping an eye on this diagnostic.