The even-parameter hook bias conjecture for 2-regular partitions

Let $b_{t,i}(n)$ denote the number of hooks of length $i$ in all $t$-regular partitions of $n$. Even-parameter hook bias conjecture. For even $k\ge 8$, $b_{2,k}(n)\ge b_{2,k+1}(n)$ for all $n\ge 0$ and $n\ne k+1$.

This is a modification of Singh and Barman's conjecture that excludes odd parameters, for which the original assertion fails. The paper reports verification for $8\le k\le 20$ and $n\le 10000$, but the general conjecture remains open.