Martino's conjecture on Calogero–Moser and Rouquier families

Let $W$ be a complex reflection group, let ${\mathcal{C}}$ be its parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ identify Calogero–Moser parameters with Hecke parameters, with $k^\sharp$ the corresponding Rouquier parameter. Martino's conjecture. For every $c\in{\mathcal{C}}$, if $k=\kappa(c)\in{\mathcal{K}}={\mathcal{C}}$, then every Calogero–Moser $c$-family is a union of Rouquier $k^\sharp$-families. The paper reports that this conjecture is confirmed in all cases computed, while the general statement is presented as a conjecture.