  A generalization of the Aubry-Andr\'e model, the non-interacting GPD model
introduced in S. Ganeshan et al.,[ Phys. Rev. Lett. 114, 146601 (2015)], is
known analytically to possess a mobility edge, allowing both extended and
localized eigenstates to coexist. This mobility edge has been hypothesized to
survive in closed many-body interacting systems, giving rise to a new
non-ergodic metallic phase. In this work, coupling the interacting GPD model to
a thermal bath, we provide direct numerical evidence for multiple qualitative
behaviors in the parameter space of disorder strength and energy level. In
particular, we look at the bath-induced saturation of entanglement entropy to
classify three behaviors: thermalized, non-ergodic extended, and localized. We
also extract the localization length in the localized phase using the long-time
dynamics of the entanglement entropy and the spin imbalance. Our work
demonstrates the rich localization landscape of generalized Aubry-Andr\'e
models containing mobility edges in contrast to the simple Aubry-Andr\'e model
with no mobility edge.
