Gordon–Martino conjecture for Calogero–Moser and Rouquier families

Let $W$ be a complex reflection group, let ${\mathcal{C}}$ be its Calogero–Moser parameter space, and let $\kappa:{\mathcal{C}}\rightarrow{\mathcal{K}}$ be the parameter correspondence, with $k^\sharp$ the associated Rouquier parameter. Gordon–Martino conjecture. If $c\in{\mathcal{C}}$ and $k=\kappa(c)\in{\mathcal{K}}={\mathcal{C}}$, then every Calogero–Moser $c$-family is a union of Rouquier $k^\sharp$-families. The paper subsequently verifies this assertion in all cases covered by its computations, but the supplied text does not establish it universally.