  Some quantum-gravity theories suggest that the absorbing horizon of a
classical black hole should be replaced by a reflective surface which is
located a microscopic distance above the would-be classical horizon. Instead of
an absorbing black hole, the resulting horizonless spacetime describes a
reflective exotic compact object. Motivated by this intriguing prediction, in
the present paper we explore the physical properties of exotic compact objects
which are linearly coupled to stationary bound-state massive scalar field
configurations. In particular, solving the Klein-Gordon wave equation for a
stationary scalar field of proper mass $\mu$ and spheroidal harmonic indices
$(l,m)$ in the background of a rapidly-rotating exotic compact object of mass
$M$ and angular momentum $J=Ma$, we derive a compact analytical formula for the
{\it discrete} radii $\{r_{\text{c}}(\mu,l,m,M,a;n)\}$ of the exotic compact
objects which can support the stationary bound-state massive scalar field
configurations. We confirm our analytical results by direct numerical
computations.
