The hook-number ratio conjecture for self-conjugate and distinct-odd partitions

Let $a_h^*(n)$ and $b_h^*(n)$ denote the numbers of hooks of length $h$ among self-conjugate partitions of $n$ and partitions of $n$ into distinct odd parts, respectively. The hook-number ratio conjecture. For every integer $h\geq2$, there exists a constant $\gamma_h^*>1$ such that

$$\frac{a_h^*(n)}{b_h^*(n)}\longrightarrow\gamma_h^*\qquad\text{as }n\to\infty.$$

This extends the comparison of hook statistics to two equinumerous families of partitions. The source presents it as an unresolved conjecture and notes that further questions about the behavior of $\gamma_h^*$ are not addressed.