  We study the thermalization of a small $XX$ chain coupled to long, gapped
$XXZ$ leads at either side by observing the relaxation dynamics of the whole
system. Using extensive tensor network simulations, we show that such systems,
although not integrable, appear to show either extremely slow thermalization or
even lack thereof since the two can not be distinguished within the accuracy of
our numerics. We show that the persistent oscillations observed in the spin
current in the middle of the $XX$ chain are related to eigenstates of the
entire system located within the gap of the boundary chains. We find from exact
diagonalization that some of these states remain strictly localized within the
$XX$ chain and do not hybridize with the rest of the system. The frequencies of
the persistent oscillations determined by numerical simulations of dynamics
match the energy differences between these states exactly. This has important
implications for open systems, where the strongly interacting leads are often
assumed to thermalize the central system. Our results suggest that if we employ
gapped systems for the leads, this assumption does not hold; this finding is
particularly relevant to any potential future experimental studies of open
quantum systems.
