Lehrer–Isaacs conjecture on character-degree polynomials for unitriangular groups

Let $U_n(q)$ be the group of unipotent upper-triangular $n\times n$ matrices over the finite field $\mathbb{F}_q$. For fixed $n$ and $e$, let $N_{n,e}(q)$ denote the number of irreducible characters of $U_n(q)$ of degree $q^e$.

Lehrer–Isaacs conjecture. For fixed $n$ and $e$, $N_{n,e}(q)$ can be expressed as a polynomial in $q$ with integer coefficients.

This refines Higman's conjecture on the polynomiality of the number of conjugacy classes of $U_n(q)$. It concerns the distribution of irreducible character degrees and is presented in the source as a conjecture attributed to Lehrer and Isaacs.