Even-hook divisibility conjecture for self-conjugate partitions

For $n\geq 0$, let $a_t^*(n)$ denote the total number of hooks of length $t$ in all self-conjugate partitions of $n$.

Even-hook divisibility conjecture. For all integers $n\geq 0$ and $m\geq 1$,

$$a_{2m}^*(n)\equiv 0\pmod{2m}.$$

The congruence strengthens the immediately preceding parity observation for even hook lengths. It was discovered experimentally in the course of studying the preceding hook-bias conjecture, and no proof or resolution is supplied in the given text.