The real-rank-zero criterion for crossed products by uniquely ergodic minimal diffeomorphisms

Let $M$ be a connected compact smooth manifold with $\dim(M)>0$, and let $h\colon M\to M$ be a uniquely ergodic minimal diffeomorphism. Let $\tau\colon C^*(\mathbf{Z},M,h)\to\mathbf{C}$ be the trace induced by the unique invariant probability measure. Real-rank-zero conjecture. The crossed-product C*-algebra $C^*(\mathbf{Z},M,h)$ has real rank zero if and only if

$$\tau_*(K_0(C^*(\mathbf{Z},M,h)))$$

is dense in $\mathbf{R}$. This would give a criterion for real rank zero in the setting of simple direct limits of recursive subhomogeneous C*-algebras with no dimension growth; the corresponding criterion for direct limits of trivial homogeneous C*-algebras is known to fail in greater generality. The status of the criterion beyond the cases discussed in the source is not resolved.