  In this short article we propose a full large $N$ asymptotic expansion of the
probability that the $m^{\text{th}}$ power of a random unitary matrix of size
$N$ has all its eigenvalues in a given arc-interval centered in $1$ when $N$ is
large. This corresponds to the asymptotic expansion of a Toeplitz determinant
whose symbol is the indicator function of several intervals having a discrete
rotational symmetry. This solves and improves a conjecture left opened by the
author. It also provides a rare example of the explicit computation of a full
asymptotic expansion of a genus $g>0$ classical spectral curve, including the
oscillating non-perturbative terms, using the topological recursion.
