The 45312-avoidance conjecture for reversal factorizations of Kazhdan–Lusztig basis elements

Let $w$ be a permutation in $\mathfrak{S}_n$. Say that $w$ avoids the pattern $45312$ if no subsequence of entries of $w$ has the same relative order as $45312$. Let $\widetilde{C}_{w}(q)$ denote the Kazhdan–Lusztig basis element associated with $w$.

45312-avoidance conjecture. If $w \in \mathfrak{S}_n$ does not avoid the pattern $45312$, then $\widetilde{C}_{w}(q)$ has no reversal factorization.

This conjecture proposes a partial answer to a question about which Kazhdan–Lusztig basis elements admit reversal factorizations, motivated by computational experimentation. The paper proves the analogous conclusion under the stronger hypotheses that $w$ avoids $4231$, does not avoid $3412$, and has $\operatorname{gap}_{3412}(w)>1$; the general claim remains open.