Marietti's invariance conjecture for parabolic Kazhdan–Lusztig polynomials

Let $(W,S)$ and $(W',S')$ be Coxeter systems, with $J\subseteq S$ and $J'\subseteq S'$. For $u,v\in W^J$ and $u',v'\in (W')^{J'}$ with $u\leqslant v$ and $u'\leqslant v'$, write

$$[u,v]^J=\{z\in W^J:u\leqslant z\leqslant v\}\subseteq [u,v]=\{z\in W:u\leqslant z\leqslant v\},$$

and similarly for $[u',v']^{J'}\subseteq [u',v']$. Here $P^{J,x}_{u,v}$ denotes the parabolic Kazhdan–Lusztig polynomial of type $x\in\{-1,q\}$. Marietti's conjecture. If there exists a poset isomorphism $[u,v]^J\simeq [u',v']^{J'}$ which extends to an isomorphism $[u,v]\simeq [u',v']$, then

$$P^{J,x}_{u,v}=P^{J',x}_{u',v'}$$

for all $x\in\{-1,q\}$. This conjecture was proposed by Marietti and proved for lower Bruhat intervals; its restriction to $J=\varnothing$ is the usual invariance conjecture for Kazhdan–Lusztig polynomials, while the general statement is the subject of the paper's equivalence results.