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.
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