Normal limits for nilpotent growth distributions

Characterize the conditions under which suitably normalized growth distributions of sequences of nilpotent groups converge to a normal distribution, including whether this holds for U(n,Z/mZ).

Background

Although the authors observe approximately normal distributions for higher-dimensional modular Heisenberg groups, they explicitly state uncertainty about the corresponding behavior of unitriangular groups. They then pose a concrete research direction concerning normal convergence for nilpotent-group growth distributions.

References

It is unclear whether this is also the case for $U(n,Z/mZ)$, which poses the following research direction: under what conditions do the (suitably normalized) growth distributions of a sequence of nilpotent groups converge to a normal distribution?

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 10, subsection “Distribution of distances in finite nilpotent groups”