Ballantine--Burson--Craig--Folsom--Wen hook length bias conjecture for self-conjugate partitions

For each $n$, let $\mathcal{SC}(n)$ be the set of self-conjugate partitions of $n$, let $\mathcal{DO}(n)$ be the set of partitions of $n$ with distinct odd parts, and let $n_t(\lambda)$ denote the number of hooks of length $t$ in a partition $\lambda$. Define

$$a_t^*(n):=\sum_{\lambda\in\mathcal{SC}(n)}n_t(\lambda),\qquad b_t^*(n):=\sum_{\lambda\in\mathcal{DO}(n)}n_t(\lambda).$$

Ballantine--Burson--Craig--Folsom--Wen hook length bias conjecture. For every integer $t\geq 2$, both of the following hold:

1. There exists an integer $N_t^*$ such that $a_t^*(n)\geq b_t^*(n)$ for all $n>N_t^*$.
2. There exists a constant $\gamma_t^*>1$ such that

$$\frac{a_t^*(n)}{b_t^*(n)}\longrightarrow\gamma_t^* \quad\text{as }n\longrightarrow\infty.$$

The conjecture formulates the eventual hook-length bias between self-conjugate partitions and partitions with distinct odd parts; the source says that Craig, Dawsey, and Han proved the corresponding conjecture, so both assertions are solved.