Gordon–Martino conjecture for Coxeter groups

Let $W$ be a finite Coxeter group with parameter $k$. Let ${\mathrm{Fam}}_k^{\mathrm{CM}}(W)$ denote the Calogero–Moser $k$-families of irreducible characters of $W$, and let ${\mathrm{Fam}}_k^{\mathrm{Lus}}(W)$ denote the Lusztig $k$-families defined from $k$-constructible characters.

Gordon–Martino conjecture. If $W$ is a Coxeter group, then

$${\mathrm{Fam}}_k^{\mathrm{CM}}(W)={\mathrm{Fam}}_k^{\mathrm{Lus}}(W).$$

This is a more precise Coxeter-group form of the Calogero–Moser/Hecke-family comparison. The equality is supported by structural results, but is not established for all Coxeter groups and parameters.