Lusztig–Dyer Combinatorial Invariance Conjecture

Let $(W,S)$ and $(W',S')$ be Coxeter systems with Bruhat orders $\leq$ and $\leq'$, respectively. For $x,y i W$ and $x',y' i W'$, suppose that the intervals $([x,y],\leq)$ and $([x',y'],\leq')$ are isomorphic as posets. Their associated Kazhdan–Lusztig polynomials are denoted by $P_{x,y}(q)$ and $P_{x',y'}(q)$.

Combinatorial Invariance Conjecture. If two Bruhat intervals, potentially from different Coxeter groups, are isomorphic as posets, then

$$P_{x,y}(q)=P_{x',y'}(q).$$

The conjecture was introduced independently by Lusztig and Dyer in the 1980s. The paper proves it for thick intervals in type $\widetilde{A}_2$; its general status is not resolved in the supplied text.