Matui's AH conjecture for effective minimal ample groupoids

Let $\mathcal{G}$ be an effective minimal second countable Hausdorff ample groupoid whose unit space $\mathcal{G}^{(0)}$ is a Cantor space. Write $H_0(\mathcal{G})$ and $H_1(\mathcal{G})$ for its first two homology groups, \left\llbracket \mathcal{G} \right\rrbracket_{\mathrm{ab}} for the abelianization of its topological full group, and let $j$ and $I_{\mathrm{ab}}$ denote the maps in the sequence below. Matui's AH conjecture. The sequence

H_0(\mathcal{G}) \otimes \mathbb{Z}_2 \xrightarrow{j} \left\llbracket \mathcal{G} \right\rrbracket_{\mathrm{ab}} \xrightarrow{I_{\mathrm{ab}}} H_1(\mathcal{G}) \longrightarrow 0

is exact. The paper proves this conjecture for Katsura--Exel--Pardo groupoids, so the general statement remains the conjectural framework beyond the cases established there.