Polynomiality conjecture for unitriangular conjugacy classes

Let $q$ be a prime power, let $m$ be a positive integer, and let $\mathbf l=(1^m)=(1,1,\ldots,1)$. With $P^{\mathbf l}(q)$ the corresponding parabolic subgroup and $N^{\mathbf l}(q)$ its nilpotent matrices, define

$$\rho_{(1^m)}(q)=\gamma(P^{(1^m)}(q),N^{(1^m)}(q)).$$

Unitriangular polynomiality conjecture. For every positive integer $m$, $\rho_{(1^m)}(q)$ is a polynomial in $q$ with rational coefficients. This is the special case corresponding to conjugacy classes of upper unitriangular matrices, and the paper notes that it is equivalent to the conjecture that their number is polynomial in $q$; its status was stated as unknown.