Odd-versus-distinct partition hook bias conjecture

Let $\mathcal O(n)$ be the set of partitions of $n$ into odd parts and let $\mathcal D(n)$ be the set of partitions of $n$ into distinct parts. For $t\geq 1$, let $a_t(n)$ and $b_t(n)$ denote the total numbers of hooks of length $t$ in the partitions in $\mathcal O(n)$ and $\mathcal D(n)$, respectively.

Odd-versus-distinct hook bias conjecture. For every integer $t\geq 2$, there exists an integer $N_t$ such that for all $n>N_t$, $a_t(n)\geq b_t(n)$. The conjectured values are

$$(N_2,N_3,N_4,N_5,N_6,N_7,N_8,N_9,N_{10})=(0,7,8,18,16,34,34,56,59).$$

Moreover, for every integer $t\geq 2$,

$$a_t(n)-b_t(n)\to\infty\quad\text{as }n\to\infty.$$

This reverses the bias at hook length $1$, where the corresponding inequality goes in the opposite direction. The assertion is proved for $t=2$ and $t=3$ in the cited paper, while the general case remains conjectural.