The AH conjecture for topological full groups of minimal étale groupoids

Let $\mathcal{G}$ be an essentially principal minimal étale groupoid whose unit space $\mathcal{G}^{(0)}$ is a Cantor set. Write $[[\mathcal{G}]]$ for its topological full group and $[[\mathcal{G}]]_{\mathrm{ab}}$ for its abelianization.

AH conjecture. There exists an exact sequence

$$H_0(\mathcal{G})\otimes\mathbb{Z}_2\longrightarrow [[\mathcal{G}]]_{\mathrm{ab}}\xrightarrow{I}H_1(\mathcal{G})\longrightarrow 0.$$

In particular, if $H_0(\mathcal{G})$ is $2$-divisible, then

$$[[\mathcal{G}]]_{\mathrm{ab}}\cong H_1(\mathcal{G}).$$

This conjecture relates the abelianization of the topological full group to the first two homology groups. The paper verifies it for many examples, but the supplied text does not establish it for all minimal essentially principal étale groupoids with Cantor unit space.