Refined degree conjecture for the polynomial class-counting function

Prove that, for every positive integer n, the polynomial k(U_n(\mathbb{F}_q)) has degree \lfloor n(n+6)/12\rfloor in q.

Background

The paper describes a refinement of Higman’s polynomiality conjecture. This refinement specifies the exact degree in q that the conjectured polynomial should have for each n. The degree values are compared with numerical estimates in Figure 1, but the conjecture remains unresolved in the paper.

References

A refinement of Higman's conjecture ( and Section 10.4) postulates that for every $n\in\mathbb{N}{*}$, $k(U_n(\mathbb{F}_q))$ is a polynomial in $q$ of degree $\lfloor n(n+6)\slash 12\rfloor$.

— Counting the number of group orbits by marrying the Burnside process with importance sampling  (2501.11731 - Diaconis et al., 20 Jan 2025) in Section 4.4, Numerical results