Accessibility-class regularity conjecture for partially hyperbolic diffeomorphisms

Let $M$ be a smooth manifold and let $f\colon M\to M$ be a $C^r$, partially hyperbolic, $r$-bunched diffeomorphism. An accessibility class for $f$ is the set of points reachable from one another by piecewise paths tangent to the stable and unstable distributions of $f$.

Accessibility-class regularity conjecture. Every accessibility class for $f$ is an injectively immersed, $C^r$ submanifold of $M$.

This conjecture would imply the saturated section theorem used in the paper, concerning smoothness of continuous sections whose images are accessibility classes for lifted fibered partially hyperbolic maps. The supplied text does not state whether the conjecture is open or resolved.