The braid-class ratio conjecture for reduced words

Let $d53e$ be the symmetric group on $n$ letters. For $d70e i d53e$, let $d4a1(d70e)$ be the set of reduced words of $d70e$ and let $d49e(d70e)$ be the set of braid classes of reduced words. The braid-class ratio conjecture. For all $d70e i d53e$,

$$\dfrac{1}{2} |\mathcal{R}(\sigma)| \leq |\mathcal{B}(\sigma)|\leq |\mathcal{R}(\sigma)|.$$

The conjecture was motivated by computations for $n=4,5,6,7,8$ and concerns the proportion of reduced words identified by braid moves; its status is not resolved in the supplied text.