  The universal relaxation bound suggests that the relaxation times of
perturbed thermodynamical systems is bounded from below by the simple
time-times-temperature (TTT) quantum relation $\tau\times T\geq
{{\hbar}\over{\pi}}$. It is known that some perturbation modes of near-extremal
Kerr black holes in the regime $MT_{\text{BH}}/\hbar\ll m^{-2}$ are
characterized by normalized relaxation times $\pi\tau\times
T_{\text{BH}}/\hbar$ which, in the approach to the limit
$MT_{\text{BH}}/\hbar\to0$, make infinitely many oscillations with a tiny
constant amplitude around $1$ and therefore cannot be used directly to verify
the validity of the TTT bound in the entire parameter space of the black-hole
spacetime (Here $\{T_{\text{BH}},M\}$ are respectively the Bekenstein-Hawking
temperature and the mass of the black hole, and $m$ is the azimuthal harmonic
index of the linearized perturbation mode). In the present compact paper we
explicitly prove that all rapidly-spinning Kerr black holes respect the TTT
relaxation bound. In particular, using analytical techniques, it is proved that
all black-hole perturbation modes in the complementary regime $m^{-1}\ll
MT_{\text{BH}}/\hbar\ll1$ are characterized by relaxation times with the simple
dimensionless property $\pi\tau\times T_{\text{BH}}/\hbar\geq1$.
