The combinatorial Sylvester four-point conjecture

Fix $n\geq 4$. Let $\mathsf{w}$ be a uniformly random reduced expression for the long word in $S_n$, and choose uniformly at random a $4$-subset of $\{1,\ldots,n\}$. The induced reduced expression $\mathsf{v}$ for the long word in $S_4$ is compared with

$$X=\{s_1s_2s_3s_2s_1s_2,\ s_3s_2s_1s_2s_3s_2,\ s_2s_1s_2s_3s_2s_1,\ s_2s_3s_2s_1s_2s_3\}.$$

Combinatorial Sylvester four-point conjecture. For every $n\geq 4$, the probability that $\mathsf{v}\in X$ is $1/4$.

This is a combinatorial analogue of Sylvester's four-point problem, obtained from the Goodman–Pollack classification of point configurations. A more general result of O. Angel and A. E. Holroyd is stated to imply this conjecture.