  We extend the Ori-Thorne-Kesden procedure to consistently describe the
non-quasi-circular transition around the ISCO from inspiral to plunge into a
black hole of arbitrary spin, including near-extremal. We identify that for
moderate or high spins the transition is governed by the Painlev\'e
transcendent equation of the first kind while for extremely high spins it is
governed by a self-similar solution to the Korteweg-de Vries equation. We match
the transition solution at leading order in the high spin limit with the
analytical quasi-circular inspiral in the near-horizon region. We also show
that the central black hole of an extreme mass ratio binary has a
near-extremality parameter that scales at least as the mass ratio due to
superradiant gravitational wave emission, which excludes extremely high spins.
