  Sorce and Wald proposed a new version of gedanken experiments to examine the
weak cosmic censorship conjecture (WCCC) in Kerr-Newmann black holes. However,
their discussion only includes the second-order approximation of perturbation
and there exists an optimal condition such that the validity of the WCCC is
determined by the higher-order approximations. Therefore, in this paper, we
extended their discussions into the high-order approximations to study the WCCC
in a nearly extremal Kerr black hole. After assuming that the spacetime
satisfies the stability condition and the perturbation matter fields satisfy
the null energy condition, based on the Noether charge method by Iyer and Wald,
we completely calculate the first four order perturbation inequalities and
discuss the corresponding gedanken experiment to overspin the Kerr black hole.
As a result, we find that the nearly extremal Kerr black holes cannot be
destroyed under the fourth-order approximation of perturbation. Then, by using
the mathematical induction, we strictly prove the $n$th order perturbation
inequality when the first $(n-1)$ order perturbation inequalities are
saturated. Using these results, we discuss the first $100$ order approximation
of the gedanken experiments and find that the WCCC in Kerr black hole is valid
under the higher-order approximation of perturbation. Our investigation implies
that the WCCC might be strictly satisfied in Kerr black holes under the
perturbation level.
