Kazhdan–Lusztig's combinatorial invariance conjecture for Kazhdan–Lusztig polynomials

Let $(W,S)$ be a Coxeter system, and let $u,v\in W$. The interval $[u,v]$ is the set of elements between $u$ and $v$ in the Bruhat order, and $P_{u,v}(q)$ denotes the classical Kazhdan–Lusztig polynomial.

Kazhdan–Lusztig's combinatorial invariance conjecture. The polynomial $P_{u,v}(q)$ depends only on the isomorphism class of the interval $[u,v]$ as a poset.

This conjecture asks whether the classical Kazhdan–Lusztig polynomial is determined by the abstract Bruhat interval, rather than by the Coxeter system and its labeling. It was independently formulated by Lusztig and Dyer; the supplied text gives no resolution status.