Ballantine–Burson–Craig–Folsom–Wen divisibility conjecture for self-conjugate partition hook lengths

Let $a_t^{\star}(n)$ denote the number of hooks of length $t$ among all self-conjugate partitions of $n$. For integers $n\geq0$ and $m\geq1$, define divisibility in the usual sense. Ballantine–Burson–Craig–Folsom–Wen conjecture. For all integers $n\geq0$ and $m\geq1$, we have

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

This is one of three conjectures of Ballantine, Burson, Craig, Folsom and Wen concerning the hook-length statistic $a_t^{\star}(n)$ for self-conjugate partitions. The supplied source does not state whether this divisibility conjecture has been resolved.