The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the diameter of this graph is known.

A computer simulation gives the conjecture that the diameter is $\lfloor n^2/4 \rfloor$. In (Heydemann, "Cayley graphs and interconnection networks", in Graph Symmetry: Algebraic Methods and Applications, Eds. Hahn and Sabidussi), it is stated in p. 213 that the diameter of the modified bubble-sort graph is $\lfloor n^2/4 \rfloor$, and the result is attributed to (J.-F. Sacle,"Diameter of some Cayley graphs", private communication, 1997). However, I couldn't find any proof of this result. Is any proof for this formula given in the literature?

This problem was studied earlier in a paper of Jerrum's ("The complexity of finding minimum-length generator sequences, TCS, 1985), but no closed-form expression for the diameter is given there either.