Cellwise Kazhdan–Lusztig classification at roots of the Poincaré polynomial

Let $\tilde{W}$ be an affine or extended affine Weyl group for which Lusztig's denominator conjecture holds, and let $P_W$ be the Poincaré polynomial of its finite Weyl group. Fix a root $q\in\mathbb{C}^{\times}$ of $P_W$, and let $\mathbf{c}$ be a two-sided cell such that, for every $w\in\mathbf{c}$ and every $x\in\tilde{W}$, the coefficient $a_{x,w}$ in Lusztig's expansion has no pole at $\mathbf{q}=q$. Let $u=u(\mathbf{c})$, and let $K(u,s,\rho,q)$ be the corresponding standard module, with simple $J$-module $E$ and $a$-value $a(E)$. Cellwise classification conjecture. Every standard module $K(u,s,\rho,q)$ has a unique simple quotient $L=L(u,s,\rho,q)$ satisfying $a(L)=a(E)$, and two such simple modules are isomorphic if and only if their corresponding triples are conjugate. This conjecture predicts that the Kazhdan–Lusztig classification survives at roots of $P_W$ for cells whose denominator coefficients remain regular there; the theorem quoted in the paper shows instead that the classification fails for the lowest cell, while the conjectural regular-cell statement remains open.