Matui's HK conjecture for ample groupoids

Let $\mathcal{G}$ be a minimal, essentially principal, ample groupoid with compact unit space. Here $K_*$ denotes $K$-theory, $C^*_r(\mathcal{G})$ denotes the reduced groupoid $C^*$-algebra, and $H_*(\mathcal{G};\mathbb{Z})$ denotes groupoid homology.

HK conjecture. For $* = 0,1$, there is an isomorphism

$$K_*\bigl(C^*_r(\mathcal{G})\bigr) \cong \bigoplus_{i = 0}^{\infty} H_{2i + *}(\mathcal{G}; \mathbb{Z}).$$

The conjecture proposes that the $K$-theory of the reduced groupoid $C^*$-algebra is determined by the even and odd groupoid homology groups. It is refuted: counterexamples were constructed by Scarparo and Deeley.