Product separability for special cube complexes

We prove that if a group $G$ admits a virtually special action on a CAT(0) cube complex, then any product of convex-cocompact subgroups of $G$ is separable. Previously, this was only known for products of three subgroups, or in the case where $G$ is hyperbolic, or in some other more technical cases with additional assumptions on the subgroups (plus these previous results assume that the action of $G$ is cocompact). We also provide an application to the action of a virtually special cubulated group on its contact graph (and to some other actions of cubulated groups on graphs).