Sharpness of the lower bound for unitriangular conjugacy classes

Establish that the lower bound q^{n^2/12(1+o_n(1))} for k(U_n(\mathbb{F}_q)) is asymptotically sharp as n tends to infinity.

Background

The paper states Higman’s lower and upper asymptotic bounds for the number of conjugacy classes of U_n(\mathbb{F}_q). It then reports Soffer’s conjecture that the lower exponent n2/12 gives the correct asymptotic growth rate. The numerical experiments later in the paper provide evidence for this conjecture, but do not prove it.

References

Soffer conjectured that the lower bound in (\ref{Bound_Higman}) is sharp.

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