Extremal Sylow p-subgroups conjecture for p-singular families of general linear groups

Let $p\in\mathbb{N}$ be prime, let $q$ be a power of $p$, and let $\mathrm{GL}_{n,q}$ denote the general linear group over the field with $q$ elements. A family is $p$-singular if its elements have order divisible by $p$. A Sylow $p$-subgroup is a subgroup of maximal order among the $p$-subgroups, and a double-translate is obtained by multiplying it on the left and right by fixed elements of $\mathrm{GL}_{n,q}$. Extremal Sylow $p$-subgroups conjecture. For all $n\geq 2$, the Sylow $p$-subgroups and their double-translates are the extremal $p$-singular families of $\mathrm{GL}_{n,q}$. The conjecture proposes the analogue of the symmetric-group extremal result for finite general linear groups; the supplied text gives no resolution.