An existing question asks whether "almost every" two-dimensional billiard possesses at least one orbit that is dense in its interior. My question is about the following set of strong counter-examples: two-dimensional billiards that contain a pair of open regions that are incompatible, in that no orbit intersects both regions. Penrose's unilluminable room is one example: an orbit that passes through the upper half of the room cannot pass through either of the square culs-de-sac in the bottom half. That example seems to rely on non-generic properties of the boundary shape (e.g., the inclusion of elliptical arcs with precisely placed foci), but perhaps this intuition is wrong.

How generic is the property of having incompatible regions? Are there known examples of billiards where this property is robust under arbitrary small perturbations of the boundary?