The divisibility conjecture for even hook numbers in self-conjugate partitions

Let $a_h^*(n)$ denote the number of hooks of length $h$ among self-conjugate partitions of $n$. The even-hook divisibility conjecture. For every integer $n\geq0$ and every integer $m\geq1$,

$$a_{2m}^*(n)\equiv0\pmod{2m}.$$

This predicts systematic divisibility for the even-hook counting functions of self-conjugate partitions. The source restates it as a conjecture from prior work, and no resolution is given in the supplied text.