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Extreme mass-ratio inspirals (EMRIs) detectable by the Laser InterferometerSpace Antenna are unique probes of the nature of supermassive compact objects.We compute the gravitational-wave signal emitted by a stellar-mass compactobject in a circular equatorial orbit around a Kerr-like horizonlesssupermassive object defined by an effective radius and a reflectivitycoefficient. The Teukolsky equations are solved consistently with suitable(frequency-dependent) boundary conditions, and the modified energy andangular-momentum fluxes are used to evolve the orbital parametersadiabatically. The gravitational fluxes have resonances corresponding to thelow-frequency quasinormal modes of the central object, which can contributesignificantly to the gravitational-wave phase. Overall, the absence of aclassical event horizon in the central object can affect the gravitational-wavesignal dramatically, with deviations even larger than those previouslyestimated by a model-independent analysis of the tidal heating. We estimatethat EMRIs could potentially place the most stringent constraint on thereflectivity of supermassive compact objects at the remarkable level of ${\calO}(10^{-6})\%$ and would allow to constrain various models which are not ruledout by the ergoregion instability. In particular, an EMRI detection could allowone to rule out (or provide evidence for) signatures of quantum black-holehorizons with Boltzmann reflectivity. Our results provide motivation forperforming rigorous parameter estimation to assess the detectability of theseeffects.

Extreme mass-ratio inspirals around a spinning horizonless compact object