Non-polynomiality conjecture for conjugacy classes of unitriangular groups

Let $U_n(q)$ be the group of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$ with ones on the diagonal, and let $k(U_n(q))$ denote its number of conjugacy classes. Non-polynomiality conjecture. The number of conjugacy classes $k(U_n(q))$ is not polynomial for $n\geq 59$. This is proposed as evidence against Higman's conjecture and is motivated by an explicitly constructed pattern subgroup embedded into $U_{59}(q)$; the source says that testing $k(U_{59}(q))$ computationally is infeasible.