Singh–Barman's 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 $n$. Singh–Barman's conjecture. For every integer $k 3$, $b_{2,k}(n) b_{2,k+1}(n)$ for all $n 0$ and $ne k+1$.

This conjecture concerns the relative frequency of consecutive hook lengths in 2-regular partitions. It is false for odd $k$: the paper gives a counterexample at $k=3$, $n=82$, and proves that every odd $ke 3$ has infinitely many counterexamples. It is proved for $k=4$ and $k=6$, while the even cases beyond these remain open.