The smoothness conjecture on orbit structure in Case 4

Let $G$ be a group generated by $C^{2}$ homeomorphisms. For a point in Case 4, consider its orbit and an interval $I$ together with subintervals $J_n$.

Smoothness conjecture. Orbits of points in Case 4 will consist of a countable number of copies of an interval $I$, the union of a number of subintervals $J_n$, in each of which the orbit is dense, has integer type, or is contained in a Cantor set.

This conjecture proposes that $C^{2}$ regularity is sufficient to obtain the expected nested description of Case 4 orbits, extending the structure known under stronger regularity assumptions. The source does not state whether it has been resolved.