  We investigate the possibility of a many-body mobility edge in the
generalized Aubry-Andr\'e (GAA) model with interactions using the Shift-Invert
Matrix Product States (SIMPS) algorithm [Phys. Rev. Lett. 118, 017201 (2017)].
The non-interacting GAA model is a one-dimensional quasiperiodic model with a
self-duality-induced mobility edge. To search for a many-body mobility edge in
the interacting case, we exploit the advantages of SIMPS that it targets
many-body states in an energy-resolved fashion and does not require all
many-body states to be localized for some to converge. Our analysis indicates
that the targeted states in the presence of the single-particle mobility edge
match neither `MBL-like' fully-converged localized states nor the fully
delocalized case where SIMPS fails to converge. We benchmark the algorithm's
output both for parameters that give fully converged, `MBL-like' localized
states and for delocalized parameters where SIMPS fails to converge. In the
intermediate cases, where the parameters produce a single-particle mobility
edge, we find many-body states that develop entropy oscillations as a function
of cut position at larger bond dimensions. These oscillations at larger bond
dimensions, which are also found in the fully-localized benchmark but not the
fully-delocalized benchmark, occur both at the band edge and center and may
indicate convergence to a non-thermal state (either localized or critical).
