  We investigate the effect of ergodic inclusions in putative many-body
localized systems. To this end, we consider the random field Heisenberg chain,
which is many-body localized at strong disorder and we couple it to an ergodic
bubble, modeled by a random matrix Hamiltonian. Recent theoretical work
suggests that the ergodic bubble destabilizes the apparent localized phase at
intermediate disorder strength and finite sizes. We tentatively confirm this by
numerically analyzing the response of the local thermality, quantified by
one-site purities, to the insertion of the bubble. For a range of intermediate
disorder strengths, this response decays very slowly, or not at all, with
increasing distance to the bubble. This suggests that at those disorder
strengths, the system is delocalized in the thermodynamic limit. However, the
numerics is unfortunately not unambiguous and we cannot definitely rule out
artefacts.
