  Extreme mass-ratio inspirals (EMRIs) detectable by the Laser Interferometer
Space Antenna are unique probes of the nature of supermassive compact objects.
We compute the gravitational-wave signal emitted by a stellar-mass compact
object in a circular equatorial orbit around a Kerr-like horizonless
supermassive object defined by an effective radius and a reflectivity
coefficient. The Teukolsky equations are solved consistently with suitable
(frequency-dependent) boundary conditions, and the modified energy and
angular-momentum fluxes are used to evolve the orbital parameters
adiabatically. The gravitational fluxes have resonances corresponding to the
low-frequency quasinormal modes of the central object, which can contribute
significantly to the gravitational-wave phase. Overall, the absence of a
classical event horizon in the central object can affect the gravitational-wave
signal dramatically, with deviations even larger than those previously
estimated by a model-independent analysis of the tidal heating. We estimate
that EMRIs could potentially place the most stringent constraint on the
reflectivity of supermassive compact objects at the remarkable level of ${\cal
O}(10^{-6})\%$ and would allow to constrain various models which are not ruled
out by the ergoregion instability. In particular, an EMRI detection could allow
one to rule out (or provide evidence for) signatures of quantum black-hole
horizons with Boltzmann reflectivity. Our results provide motivation for
performing rigorous parameter estimation to assess the detectability of these
effects.
