The invariant-measure classification conjecture for sphere crossed products

Let $g$ and $h$ be minimal diffeomorphisms of $S^m$ and $S^n$, respectively, with $m,n\geq 3$ and odd. Let the associated transformation group C*-algebras be $C^*(\mathbf{Z},S^m,g)$ and $C^*(\mathbf{Z},S^n,h)$, and let the spaces of invariant Borel probability measures be the spaces of Borel probability measures invariant under $g$ and $h$, respectively. Invariant-measure classification conjecture. If the spaces of $g$-invariant and $h$-invariant Borel probability measures are affinely homeomorphic, then

$$C^*(\mathbf{Z},S^m,g)\cong C^*(\mathbf{Z},S^n,h).$$

This is presented as a special case of the Elliott classification conjecture, motivated by the preceding example showing that the Elliott invariant can depend only on the space of invariant Borel probability measures. The source does not provide evidence of a resolution.