Singh and Barman's hook length bias conjecture for t-regular partitions

Let $b_{t,k}(n)$ denote the number of hooks of length $k$ across all $t$-regular partitions of $n$. Singh and Barman's conjecture. For every integer $t\geq 3$ and every $n\geq 0$,

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

The conjecture compares the total number of $2$-hooks in $(t+1)$-regular and $t$-regular partitions. The paper proves this inequality, so the conjecture is resolved.