Pseudo-filling symmetry conjecture for Gasharov–Reiner permutations

Let $w$ be a permutation in the class avoiding $4231$, $35142$, $42513$, and $351624$. Let $PF^{{\rm L}}_w(q)$ and $PF^{\rotatebox[origin=c]{180}{\scriptsize {\rm L}}}_w(q)$ be the generating functions for pseudo-${\rm L}$-fillings and pseudo-$\rotatebox[origin=c]{180}{{\rm L}}$-fillings of the diagram $E_w$, respectively, and let $P_w(q)$ be the relevant Kazhdan–Lusztig polynomial. Pseudo-filling symmetry conjecture. One has

$$PF^{{\rm L}}_w(q)=PF^{\rotatebox[origin=c]{180}{\scriptsize {\rm L}}}_w(q)=q^{\ell(w)}P_w(q^{-1}).$$

This conjecture extends the stated Bruhat-skew equality from ordinary fillings to pseudo-fillings for Gasharov–Reiner permutations; the source gives no resolution.