Accessibility implies ergodicity for partially hyperbolic diffeomorphisms

Let $f$ be a $C^{2}$ partially hyperbolic, volume-preserving diffeomorphism, and suppose that $f$ has the essential accessibility property, meaning that every measurable set saturated by accessibility classes has either zero or full volume. Accessibility implies ergodicity. The diffeomorphism $f$ is ergodic. The supplied status evidence says that this conjecture was proved in full generality for arbitrary center dimension in the $C^{1}$ topology.