Elder's conjecture on one-element commutation classes

Let $n\geq 1$ and let $\sigma\in\mathfrak{S}_{n+1}$. Write $R(\sigma)$ for the set of reduced words of $\sigma$ and $C(\sigma)$ for the set of commutation classes of reduced words. Elder's conjecture. For all $\sigma\in\mathfrak{S}_{n+1}$,

$$0\leq |C(\sigma)|\leq\frac{1}{2}|R(\sigma)|+1.$$

This conjecture was proposed by Elder and concerns the number of commutation classes relative to the number of reduced words. The paper establishes a bound of four for the number of one-element commutation classes, but the conjecture itself is not resolved here.