Relative combinatorial invariance conjecture for relative R-polynomials

Let $(W,S)$ be a finite Coxeter system. For a subset $J\subseteq S$, write $W_J$ for the corresponding standard parabolic subgroup. For pairs in finite Coxeter systems, let $G[\sigma_i,\omega_i]$ be their directed Bruhat intervals and let $R_{\sigma_i,\omega_i,J}$ denote the $J$-relative $R$-polynomial. Relative combinatorial invariance conjecture. For every finite Coxeter system $(W,S)$, there is an assigned subset $\mathbb{J}(W)\subsetneq S$ such that, whenever two Bruhat intervals are isomorphic by $\phi$ and every edge $\sigma_1\leq z<tz\leq\omega_1$ satisfies

$$t\in (W_1)_{\mathbb{J}(W_1)}\quad\Longleftrightarrow\quad \phi(tz)\phi(z)^{-1}\in (W_2)_{\mathbb{J}(W_2)},$$

one has

$$R_{\sigma_1,\omega_1,\mathbb{J}(W_1)}=R_{\sigma_2,\omega_2,\mathbb{J}(W_2)}.$$

This is proposed as a relative analogue of combinatorial invariance, formulated through $J$-relative $R$-polynomials. Its status is not specified in the supplied text.