Polynomiality conjecture for nilpotent conjugacy classes in parabolic subgroups

Let $q$ be a prime power and let $\mathbf l=(l_1,\ldots,l_s)$ be a tuple of nonnegative integers. For $m=l_1+\cdots+l_s$, let $P^{\mathbf l}(q)$ be the parabolic subgroup of $\operatorname{GL}_m(q)$ consisting of the invertible matrices in the block upper-triangular matrix algebra $M^{\mathbf l}(q)$, and let $N^{\mathbf l}(q)$ be the set of nilpotent matrices in $M^{\mathbf l}(q)$. Write

$$\rho_{\mathbf l}(q)=\gamma(P^{\mathbf l}(q),N^{\mathbf l}(q)),$$

where $\gamma(G,X)$ denotes the number of orbits of the action of $G$ on the finite set $X$. Polynomiality conjecture. For every tuple $\mathbf l=(l_1,\ldots,l_s)$ of nonnegative integers, $\rho_{\mathbf l}(q)$ is polynomial in $q$, meaning that there is a polynomial with rational coefficients agreeing with it for every prime power $q$. The conjecture concerns the dependence on $q$ of nilpotent conjugacy-class counts in arbitrary parabolic subgroups; the paper states that this polynomiality was not known in general.