Conjecture on generic non-absolute continuity of central foliations

A volume-preserving partially hyperbolic diffeomorphism is a diffeomorphism preserving a volume measure and admitting a partially hyperbolic invariant splitting; when its center foliation exists, this is the foliation tangent to the center bundle. Generic central-foliation conjecture. Generically the central foliation, if it exists, of a volume-preserving partially hyperbolic diffeomorphism is non-absolutely continuous. This conjecture concerns the typical regularity of central foliations in partially hyperbolic dynamics and is motivated by examples and generic results for systems with compact center leaves; its general status is not specified in the source.