On hook length biases in $t$-regular partitions

Let $t\ge 2$ and $k\ge 1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\ge b_{2,2}(n)$ for all $n\ge 4$, and conjectured that $b_{t+1,2}(n)\ge b_{t,2}(n)$ for all $t\ge 3$ and $n\ge 0$. In this paper, we prove that the conjecture is true for $t=3$.