Higman’s polynomiality conjecture for unitriangular conjugacy classes

Prove that, for every positive integer n, the number of conjugacy classes k(U_n(\mathbb{F}_q)) of the unitriangular group U_n(\mathbb{F}_q) is a polynomial in the prime power q.

Background

The paper studies approximate counting of conjugacy classes in the unitriangular groups U_n(\mathbb{F}_q). Higman’s conjecture asserts that the resulting class-counting function, viewed as a function of the field size q for each fixed n, has polynomial rather than more complicated dependence on q. The paper cites existing work on the status of this conjecture but does not resolve it.

References

For every $n\in \mathbb{N}{*}$, $k(U_n(\mathbb{F}_q))$ is a polynomial in $q$.

— Counting the number of group orbits by marrying the Burnside process with importance sampling  (2501.11731 - Diaconis et al., 20 Jan 2025) in Conjecture in Section 1, Introduction