  It is known that a near-extremal Kerr black hole can be spun up beyond its
extremal limit by capturing a test particle. Here we show that overspinning is
always averted once back-reaction from the particle's own gravity is properly
taken into account. We focus on nonspinning, uncharged, massive particles
thrown in along the equatorial plane, and work in the first-order self-force
approximation (i.e., we include all relevant corrections to the particle's
acceleration through linear order in the ratio, assumed small, between the
particle's energy and the black hole's mass). Our calculation is a numerical
implementation of a recent analysis by two of us [Phys.\ Rev.\ D {\bf 91},
104024 (2015)], in which a necessary and sufficient "censorship" condition was
formulated for the capture scenario, involving certain self-force quantities
calculated on the one-parameter family of unstable circular geodesics in the
extremal limit. The self-force information accounts both for radiative losses
and for the finite-mass correction to the critical value of the impact
parameter. Here we obtain the required self-force data, and present strong
evidence to suggest that captured particles never drive the black hole beyond
its extremal limit. We show, however, that, within our first-order self-force
approximation, it is possible to reach the extremal limit with a suitable
choice of initial orbital parameters. To rule out such a possibility would
require (currently unavailable) information about higher-order self-force
corrections.
