Calogero–Moser and Kazhdan–Lusztig family and character conjecture

Let $W$ be a Coxeter group and let $c\in {\boldsymbol{\mathcal C}}_{\mathbb R}$. The Calogero–Moser and Kazhdan–Lusztig theories define partitions of $\operatorname{Irr}(W)$ into $c$-families, as well as sets of Calogero–Moser and Kazhdan–Lusztig $c$-cell characters.

Family and character conjecture. The partition of $\operatorname{Irr}(W)$ into Calogero–Moser $c$-families coincides with the partition into Kazhdan–Lusztig $c$-families, as conjectured by Gordon–Martino. If $c_s\geq 0$ for every $s\in\operatorname{Ref}(W)$, then the set of Calogero–Moser $c$-cell characters coincides with the set of Kazhdan–Lusztig $c$-cell characters.

These assertions are consequences at the character level of the corresponding two-sided and left-cell conjectures and remove the dependence on a choice of prime ideal. The paper presents them as conjectures; their general status is open in the supplied text.