Conjecture on the diameter of the Hurwitz graph of the symmetric group

Let $G_T(n)$ be the Hurwitz graph whose vertices are reduced factorizations of the reverse permutation in $S_n$ into transpositions, with edges given by local Hurwitz moves. Diameter conjecture. The diameter of $G_T(n)$ is

$${n-1\choose 2}+O(n).$$

The preceding proposition gives an upper bound of $\frac{3}{2}{n-1\choose 2}$, while the conjecture predicts the sharper leading term. Computations for $n<8$ suggest the more precise value $\left\lfloor (n-1)^2/2 \right\rfloor-1$, but this stronger formula is not asserted as the conjecture here.