Monotonicity conjecture for hook counts in t-regular partitions

For integers $t\geq2$ and $k\geq1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-regular partitions of $n$. A $t$-regular partition is a partition with no part divisible by $t$.

Monotonicity conjecture. Let $t\geq3$ be an integer. Then

$$b_{t+1,2}(n)\geq b_{t,2}(n),$$

for all $n\geq0$.

The conjecture formalizes numerical evidence that the number of hooks of length $2$ in $t$-regular partitions increases with $t$. The source gives no resolution, so its status remains open.