The conjecture on the possible numbers of one-element commutation classes

Let $n\geq 1$ and let $\sigma\in\mathfrak{S}_{n+1}$. Let $R_{\bullet}(\sigma)$ denote the set of one-element commutation classes of reduced words of $\sigma$. The possible-cardinalities conjecture. For all $\sigma\in\mathfrak{S}_{n+1}$,

$$|R_{\bullet}(\sigma)|\in\{0,1,2,4\}.$$

Theorem 7.1 gives the upper bound $|R_{\bullet}(\sigma)|\leq 4$, while computation through $n=8$ found no example with exactly three one-element commutation classes. The conjecture asserts that three never occurs; the source does not report a proof or disproof.