And may I just confirm that, earlier this morning you mentioned the prior is just a domain(at least for the current version), which sounds it won’t enter the posterior distribution like a real prior.
Is this the case for both FOCE and HMC-NUTS?
Yes. That is indeed the case for FOCE(I) (and LaplaceI) but for the NUTS sampler aka BayesMCMC the likelihood contribution of the prior is applied as you would expect.