Polynomiality conjecture for double cosets of parabolic linear groups

Let $n$ be a positive integer, let $\lambda,\mu\vdash n$, and let $q$ be a prime power. Write $\operatorname{GL}_n(\mathbb F_q)$ for the general linear group and $\operatorname{GL}_\lambda(\mathbb F_q)$ and $\operatorname{GL}_\mu(\mathbb F_q)$ for the subgroups associated with the partitions $\lambda$ and $\mu$. Polynomiality conjecture. The double-coset number

$$\left|\operatorname{GL}_\lambda(\mathbb F_q)\backslash\operatorname{GL}_n(\mathbb F_q)/\operatorname{GL}_\mu(\mathbb F_q)\right|$$

is a monic polynomial in $q$ with positive integer coefficients. This conjecture predicts a particularly positive enumeration formula for double cosets of the indicated subgroups of a general linear group; the source gives no resolution, so its status remains open.