Diameter at modulus two and metacyclic groups

Prove that the diameter of U(n,Z/2Z) with respect to the fundamental-root transvections equals the number of metacyclic groups of order 2^n.

Background

The authors report computational evidence for an unexpected enumerative relation between a Cayley-graph diameter and the number of metacyclic groups of order 2n. They formulate this relation as a separate conjecture.

References

The diameter of $U(n,\mathbb{Z}/2\mathbb{Z})$ with respect to the fundamental root transvections is equal to the number of metacyclic groups of order $2n$ (\href{https://oeis.org/A136184}{OEIS A136184}).

— 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”