Higman's polynomiality conjecture for unitriangular groups

Let $n\in\mathbb{N}^{*}$, let $q$ be a prime power, and let $U_n(\mathbb{F}_q)$ be the group of $n\times n$ upper triangular matrices over $\mathbb{F}_q$ with ones on the diagonal. Write $k(U_n(\mathbb{F}_q))$ for its number of conjugacy classes. Higman's conjecture. For every $n\in\mathbb{N}^{*}$, $k(U_n(\mathbb{F}_q))$ is a polynomial in $q$. This conjecture predicts uniform polynomial dependence on the field size and motivates the study of the conjugacy classes of unitriangular groups. Its status is not specified in the source.