  We study the interplay between interactions and finite-temperature dephasing
baths. We consider a double well with strongly interacting bosons coupled, via
the density, to a bosonic bath. Such a system, when the bath has infinite
temperature and instantaneous decay of correlations, relaxes with an emerging
algebraic behavior with exponent 1/2. Here we show that, because of the
finite-temperature baths and of the choice of spectral densities, such an
algebraic relaxation may occur for a shorter duration and the characteristic
exponent can be lower than 1/2. These results show that the interaction-induced
impeding of relaxation is stronger and more complex when the bath has finite
temperature and/or nonzero timescale for the decay of correlations.
