Theory question: Adaptive HMC for energy transition distribution?

Hi all,
I am in the midst of writing about HMC working from Betancourt’s “A conceptual introduction to Hamiltonian Monte Carlo” and I see that the stochastic exploration between level sets is effective if the energy transition probability matches the marginal energy distribution, which to my mind parallels how in random walk MCMC exploration is effective if the proposal distribution matches the target distribution which is what adaptive MC exploits by tailoring the proposal based on accumulated knowledge.

So I suppose either I have just stumbled into part of the current research activity around HMC or I am misinterpreting, which is why I have put my theory question here! I have spent quite a bit of time reading into HMC itself and trying to use it for my own work (but not so much time into adaptive MC beyond cursory knowledge) so I appreciate any external input on this!