Diameter of unitriangular groups over Z/mZ

Determine the eventual diameters of U(n,Z/mZ), for fixed n and sufficiently large m, with respect to fundamental-root and positive-root transvections in both undirected and oriented Cayley graphs, according to the four formulas stated in the conjecture.

Background

Computational experiments suggest that, contrary to previously known bounds involving both n and log(m), the diameters of unitriangular groups over Z/mZ become linear in m for fixed n. The paper states four explicit conjectural formulas depending on the generating set and whether inverses are allowed.

References

For large enough $m$ (depending on $n$) the diameter of $U(n, \mathbb{Z}/m\mathbb{Z})$ is a follows:

— 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 9, subsection “Bounds on the diameters of unitriangular groups”