Conjecture on the roots of the Harada invariant for general linear groups

Let $k$ be a finite field, let $n\geq 6$, and write $h(\operatorname{GL}_n(k))$ for the Harada invariant of the finite general linear group. Let $v_q(h(\operatorname{GL}_n(k)))$ denote the polynomial in the field order $q$ obtained in the paper.

Root bound conjecture. For $n\geq 6$, the maximum real root of $v_q(h(\operatorname{GL}_n(k)))$ is less than $2$.

The bound would imply that $h(\operatorname{GL}_n(k))$ is integral for every finite field with $q\geq 2$, extending the verified cases $2\leq n\leq 5$ and the theorem for sufficiently large $q$.