  We present a new computation of the asymptotic gravitational wave energy
fluxes emitted by a {\it spinning} particle in circular equatorial orbits about
a Kerr black hole. The particle dynamics is computed in the pole-dipole
approximation, solving the Mathisson-Papapetrou equations with the Tulczyjew
spin-supplementary-condition. The fluxes are computed, for the first time, by
solving the 2+1 Teukolsky equation in the time-domain using hyperboloidal and
horizon-penetrating coordinates. Denoting by $M$ the black hole mass and by
$\mu$ the particle mass, we cover dimensionless background spins
$a/M=(0,\pm0.9)$ and dimensionless particle spins $-0.9\leq S/\mu^2 \leq +0.9$.
Our results span orbits of Boyer-Lindquist coordinate radii $4\leq r/M \leq
30$; notably, we investigate the strong-field regime, in some cases even beyond
the last-stable-orbit. We confirm, numerically, the Tanaka {\it et al.} [Phys.\
Rev.\ D 54, 3762] 2.5th order accurate Post-Newtonian (PN) predictions for the
gravitational wave fluxes of a spinning particle in circular equatorial orbit:
we find an unambiguos trend of the PN-prediction towards the numerical results
when $r$ is large. At $r/M=30$ the fractional agreement between the full
numerical flux, approximated as the sum over the modes $m=1,2,3$, and the PN
prediction is $\lesssim0.5\%$ in all cases tested. This is close to our
fractional numerical accuracy ($\sim 0.2\%$). For smaller radii, the agreement
between the 2.5PN prediction and the numerical result progressively
deteriorates. Our numerical data will be essential to develop suitably resummed
expressions of the 2.5PN analytical fluxes in order to improve their accuracy
in the strong-field regime.
