The cycle characterization conjecture for cyclically fully commutative elements in type A

Let $W(A_n)$ be the Coxeter group of type $A_n$, identified with the symmetric group $S_{n+1}$, and let $w\in W(A_n)$ correspond to a permutation with disjoint cycles $c_1,c_2,\ldots,c_k$. Write each $c_j$ with its smallest number first. A cycle has connected support if the set of numbers appearing in it consists of consecutive numbers, and it has a direction change at $w(i)$ if either $i<w(i)$ and $w(i)>w^2(i)$, or $i>w(i)$ and $w(i)<w^2(i)$. The cycle characterization conjecture. The element $w$ is cyclically fully commutative if and only if every cycle $c_j$ has connected support and has at most one direction change. This conjecture gives a direct criterion for recognizing cyclically fully commutative elements from the cycle decomposition of the corresponding permutation; the authors report computational evidence from Sage but do not establish the assertion in general.