Lusztig–Calogero–Moser cuspidal-family conjecture

Let $W$ be a finite Coxeter group, let $V_{\mathbb R}$ be a real form of its reflection representation, and let $\tau\in {\mathrm N}_{{\mathbf G}{\mathbf L}_{\mathbb R}(V_{\mathbb R})}(W)$ be $W$-full and satisfy $\tau(k)=k$. Denote by the $\tau$-cuspidal Lusztig $k$-families and $\tau$-cuspidal Calogero–Moser $k$-families the subcollections of the corresponding family partitions consisting of cuspidal families.

Cuspidal-family conjecture. The $\tau$-cuspidal Lusztig $k$-families coincide with the $\tau$-cuspidal Calogero–Moser $k$-families.

This predicts that the Gordon–Martino comparison preserves the cuspidality structure associated with a linear automorphism. Its general status is not resolved in the source.