Uniqueness and length of the longest LRX element

Prove that the permutation (0,1)(n-1,2)(n-2,3)(n-3,4)… is the unique longest element of the LRX Cayley graph of S_n and has word length n(n-1)/2.

Background

Breadth-first searches through the LRX graphs for n≤14 indicate that the longest element is unique and has a product-of-transpositions form associated with a symmetry of a regular polygon. The paper gives an explicit decomposition of the candidate with the conjectured length, but does not prove either uniqueness or its maximality for all n.

References

We conjecture that this element is unique longest element and it has length $n(n-1)/2$.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 5.2, “Long and the longest element”