HHU Problem 16 on the structure and regularity of accessibility classes

Let $M$ be a manifold and let $f:M\to M$ be a partially hyperbolic diffeomorphism. For $x\in M$, the accessibility class $AC(x)$ is the set of points that can be joined to $x$ by a path piecewise contained in leaves of the strong stable and strong unstable foliations. HHU Problem 16. Prove that the accessibility classes are topological manifolds that vary semicontinuously, as do their dimensions; with bunching, prove that they are smooth manifolds. The paper's main theorem proves that for a two-dimensional center bundle these classes are immersed manifolds, and its differentiability theorem gives injectively immersed $C^1$ submanifolds under stated bunching hypotheses, but the full problem remains open.