Two-sided Calogero–Moser and Kazhdan–Lusztig cells conjecture

Let $W$ be a Coxeter group and let $c\in {\boldsymbol{\mathcal C}}_{\mathbb R}$. Choose a prime ideal ${\bar{\mathfrak r}}_c$ above ${\bar{\mathfrak q}}_c$; this choice determines the two-sided Calogero–Moser $c$-cells. Write $\operatorname{Irr}^{\mathrm{CM}}_\Gamma(W)$ and $\operatorname{Irr}^{\mathrm{KL}}_\Gamma(W)$ for the irreducible-character sets attached to a two-sided cell $\Gamma$.

Two-sided cell conjecture. There exists a choice of ${\bar{\mathfrak r}}_c$ such that the partition of $W$ into two-sided Calogero–Moser $c$-cells coincides with the partition into two-sided Kazhdan–Lusztig $c$-cells. If $c_s\geq 0$ for every $s\in\operatorname{Ref}(W)$, then for every common two-sided cell $\Gamma$,

$$\operatorname{Irr}^{\mathrm{CM}}_\Gamma(W)=\operatorname{Irr}^{\mathrm{KL}}_\Gamma(W).$$

This conjecture refines the general comparison of the two cell theories by including the associated character families. Its status is open in the supplied text.