The E-BFMI (really should be E-FMI) quantifies how well the momentum resampling in HMC works. Low values indicate that the sampler is not able to explore all of the relevant energy sets fast enough which may manifest as bias in the exploration of the parameter space.
For manipulating the energy diagnostic take a look at https://github.com/betanalpha/knitr_case_studies/blob/master/fitting_the_cauchy/stan_utility.R. In particular, run check_energy to see exactly what the E-FMI values are – RStan warns for values below 0.3 but we have since seen that dynamic HMC works reasonably well for values as low as 0.2.
If you still see E-FMI values lower than 0.2 then you’ll have to consider reparameterizations/new priors. In particular, in many cases low E-FMI seems to indicate really poor identifiability of the likelihood which will require careful prior choice to ensure a well-behaved joint model.