On hook length biases in $t$-regular partitions

Let $t\geq2$ and $k\geq1$ 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 $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$.