The hook-number ratio conjecture for odd-part and distinct-part partitions

Let $\mathcal O(n)$ and $\mathcal D(n)$ denote the collections of partitions of $n$ into odd parts and distinct parts, respectively. For an integer $h\geq 1$, let

$$a_h(n)=\sum_{\lambda\in\mathcal O(n)}\#\{x\in\mathcal H(\lambda):x=h\},\qquad b_h(n)=\sum_{\lambda\in\mathcal D(n)}\#\{x\in\mathcal H(\lambda):x=h\}.$$

The hook-number ratio conjecture. For fixed $h\geq 2$, there exists an integer $N_h>0$ such that $a_h(n)\geq b_h(n)$ for all $n>N_h$, and 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 conjecture predicts that, although partitions into distinct parts have at least as many hooks of length $1$ as partitions into odd parts, the inequality reverses asymptotically for every fixed hook length $h\geq2$. The source paper proves the conjecture motivating these statements, but this formulation is presented as a conjecture in the cited prior work; its resolution is not specified here.