Filtered combinatorial invariance conjecture for Kazhdan–Lusztig polynomials

Let $(W,S)$ be a finite Coxeter system equipped with a canonical filtration

$$\underline{J}^0=(\emptyset=J_0\subsetneq J_1\subsetneq\cdots\subsetneq J_r=S).$$

This filtration gives a reflection-coloring map $\alpha:T\to\{1,\ldots,r\}$ by assigning to each reflection the least $i$ such that it lies in $W_{J_i}$. Filtered combinatorial invariance conjecture. For any finite Coxeter system, there is an assigned canonical filtration $\underline{J}^0$ such that, for two filtered finite Coxeter systems and an isomorphism $\phi$ of their directed Bruhat intervals, if there is an increasing map $\gamma:\{1,\ldots,r_1\}\to\{1,\ldots,r_2\}$ satisfying

$$\gamma\circ\alpha_1(t)=\alpha_2(t')$$

for every edge $\sigma_1\leq z<tz\leq\omega_1$, where $t'\in W_2$ is the reflection determined by $\phi(tz)=t'\phi(z)$, then

$$P_{\sigma_1,\omega_1}=P_{\sigma_2,\omega_2}.$$

The conjecture refines combinatorial invariance by requiring preservation of reflection colors up to an increasing relabeling. Its status is not specified in the supplied text.