The 3412-pattern upper-bound conjecture for reduced-word graphs

Let $S_n$ be the symmetric group, let $w\in S_n$, let $G(w)$ be the graph of reduced words of $w$, let $L_2(w)$ denote the associated set of rank-two root subsystems, and let $N_{3412}(w)$ be the number of occurrences of the pattern $3412$ in $w$. The 3412-pattern upper-bound conjecture. For any permutation $w\in S_n$, we have

$$\operatorname{diam}(G(w))\leq |L_2(w)|-N_{3412}(w).$$

This conjecture is based on computational evidence and strengthens conjectured upper bounds of Reiner and Roichman and of Dahlberg and Kim. Its status is not resolved in the supplied text.