  We investigate the robustness of the many-body localized (MBL) phase to the
quantum-avalanche instability by studying the dynamics of a localized spin
chain coupled to a $T=\infty$ thermal bath through its leftmost site. By
analyzing local magnetizations, we estimate the size of the thermalized sector
of the chain and find that it increases logarithmically slowly in time. This
logarithmically slow propagation of the thermalization front allows us to lower
bound the slowest thermalization time, and find a broad parameter range where
it scales fast enough with the system size that MBL is robust against
thermalization induced by avalanches. The further finding that the imbalance --
a global quantity measuring localization -- thermalizes over a time scale
exponential both in disorder strength and system size is in agreement with
these results.
