The HK conjecture for minimal étale groupoids

Let $\mathcal{G}$ be an essentially principal minimal étale groupoid whose unit space $\mathcal{G}^{(0)}$ is a Cantor set.

HK conjecture. The even and odd homology groups of $\mathcal{G}$ satisfy

$$\bigoplus_{i=0}^\infty H_{2i}(\mathcal{G})\cong K_0(C^*_r(\mathcal{G}))$$

and

$$\bigoplus_{i=0}^\infty H_{2i+1}(\mathcal{G})\cong K_1(C^*_r(\mathcal{G})).$$

This conjecture proposes a homological description of the $K$-groups of the reduced groupoid $C^*$-algebra. It is verified in the paper for many examples, including minimal $\mathbb{Z}$-actions, one-sided shifts of finite type, and products of such groupoids; its general status is not resolved by the supplied text.