Largest diameters contain involutions

Prove that, for infinitely many n, a generating set attaining the largest diameter of a Cayley graph of S_n contains an involution, in both directed and undirected cases.

Background

Exhaustive searches for small n and randomized searches for larger n found especially large diameters when one generator is an involution. The authors formulate this as a conjecture for both directed and undirected Cayley graphs.

References

The generating set with largest diameter for $S_n$ contains an involution (at least for infinite number of $n$), both for directed and undirected cases.

— 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 5, subsection “Distribution of diameter over conjugacy classes pairs -- involutions strikes”