The Calogero–Moser and Kazhdan–Lusztig two-sided-cell conjecture

Let $W$ be a finite real reflection group with parameter $c$. A Calogero–Moser two-sided $c$-cell is denoted by $\Gamma$, and let $\operatorname{Irr}_\Gamma^{c,\mathrm{CM}}(W)$ and $\operatorname{Irr}_\Gamma^{c,\mathrm{KL}}(W)$ denote the Calogero–Moser and Kazhdan–Lusztig families of irreducible representations attached to a two-sided cell. Calogero–Moser–Kazhdan–Lusztig two-sided-cell conjecture. Calogero–Moser two-sided $c$-cells coincide with Kazhdan–Lusztig two-sided $c$-cells, and, for each common cell $\Gamma$,

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

This is the two-sided analogue of the left-cell conjecture and is presented as open; the available evidence includes the corresponding results in type $A$.