Imaginary-part bound for roots of the box polynomial

Let $B_{m,n}(x)$ be the box polynomial, and consider its complex roots. The notation $O(m\cdot\sqrt{n})$ denotes an asymptotic upper bound in the parameters $m$ and $n$. Imaginary-part bound conjecture. The imaginary part of every root of $B_{m,n}(x)$ is bounded by $O(m\cdot\sqrt{n})$. The preceding theorem gives the weaker bound $O(m\cdot n)$, while computational data suggests this sharper order of growth; whether the sharper bound holds remains open.