  It is known that strong disorder in closed quantum systems leads to many-body
localization (MBL), and that this quantum phase can be destroyed by coupling to
an infinitely large Markovian environment. However, the stability of the MBL
phase is less clear when the system and environment are of finite and
comparable size. Here, we study the stability and eventual localization
properties of a disordered Heisenberg spin chain coupled to a finite
environment, and extensively explore the effects of environment disorder,
geometry, initial state and system-bath coupling strength. By studying the
non-equilibrium dynamics and the eventual steady-state properties of different
initial states, our numerical results indicate that in most cases, the system
retains its localization properties despite the coupling to the finite
environment, albeit to a reduced extent. However, in cases where the system and
environment is strongly coupled in the ladder configuration, the eventual
localization properties are highly dependent on the initial state, and could
lead to either thermalization or localization.
