Lehrer's conjecture on character-degree polynomials of unitriangular groups

Let $q$ be a power of an odd prime, let $U_n(q)$ be the unitriangular group of degree $n$ over the finite field ${\mathbb F}_q$, and let $e\in {\mathbb N}$. Lehrer's conjecture. The number of irreducible characters of $U_n(q)$ with degree $q^e$ is a polynomial in $q$ with integral coefficients. This conjecture concerns the polynomial behavior of character-degree counts for unitriangular groups; the paper studies related counting functions for unipotent radicals of maximal standard parabolic subgroups and provides explicit formulas in those settings.