Marietti's parabolic Combinatorial Invariance Conjecture

Let $(W_1,S_1)$ and $(W_2,S_2)$ be Coxeter systems, with $J_1\subseteq S_1$ and $J_2\subseteq S_2$. Let $u,v\in W_1^{J_1}$, $w,z\in W_2^{J_2}$, and let $f:[u,v]\to[w,z]$ be a poset isomorphism satisfying

$$f([u,v]^{J_1})=[w,z]^{J_2}.$$

Parabolic Combinatorial Invariance Conjecture. Then

$$P_{u,v}^{J_1,q}(q)=P_{w,z}^{J_2,q}(q).$$

The naive parabolic version is false in general, whereas this deeper formulation is known when $u=w=e$ and remains open in general.