  Many-body localization (MBL) behavior is analyzed {in an extended
Bose-Hubbard model with quasiperiodic infinite-range interactions. No
additional disorder is present. Examining level statistics and entanglement
entropy of eigenstates we show that a significant fraction of eigenstates of
the system is localized in the presence of strong interactions. In spite of
this, our results suggest that the system becomes ergodic in the standard
thermodynamic limit in which the energy of the system is extensive. At the same
time, the MBL regime seems to be stable if one allows for a super-extensive
scaling of the energy. We show that our findings can be experimentally verified
by studies of time dynamics in many-body cavity quantum electrodynamics setups.
The "quench spectroscopy" is a particularly effective tool that allows us to
systematically study energy dependence of time dynamics and to investigate a
mobility edge in our system.
