Transposon diameter conjecture

Prove that the diameter of the Cayley graph generated by block transpositions, or transposons, equals ceil((n+1)/2) for all n except n=13 and n=15.

Background

The paper presents the exact diameter formula as a conjecture attributed to prior literature. The generators model block transpositions arising in genome rearrangement problems.

References

{\bf Conjecture.}(\href{https://oeis.org/A065603}{OEIS-A065603}) Diameter of transposons is $\left\lceil \tfrac{n+1}{2}\right\rceil$ for $n\ne 13,15$.

— 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 14, subsection “Transposons (block transpositions)”