  Many interesting experimental systems, such as cavity QED or central spin
models, involve global coupling to a single harmonic mode. Out-of-equilibrium,
it remains unclear under what conditions localized phases survive such global
coupling. We study energy-dependent localization in the disordered Ising model
with transverse and longitudinal fields coupled globally to a $d$-level system
(qudit). Strikingly, we discover an inverted mobility edge, where high energy
states are localized while low energy states are delocalized. Our results are
supported by shift-and-invert eigenstate targeting and Krylov time evolution up
to $L=13$ and $18$ respectively. We argue for a critical energy of the
localization phase transition which scales as $E_c \propto L^{1/2}$, consistent
with finite size numerics. We also show evidence for a reentrant MBL phase at
even lower energies despite the presence of strong effects of the central mode
in this regime. Similar results should occur in the central spin-$S$ problem at
large $S$ and in certain models of cavity QED.
