The rational HK-conjecture for principal groupoids

Let $\mathcal{G}$ be a second countable, étale, principal, minimal, ample groupoid. Write $C^*_r(\mathcal{G})$ for its reduced groupoid $C^*$-algebra, $K_*(C^*_r(\mathcal{G}))$ for its $K$-theory, and $H_*(\mathcal{G})$ for its groupoid homology.

Rational HK-conjecture. One should have

$$K_*(C^*_r(\mathcal{G}))\otimes \mathbb{Q} \cong \left(\bigoplus_i H_{*+2i}(\mathcal{G})\right)\otimes \mathbb{Q}.$$

The rational version weakens the integral HK-conjecture by tensoring both sides with $\mathbb{Q}$. The examples constructed in the paper satisfy this rational version, while the integral conjecture fails for the principal example; its general status is therefore not resolved by the counterexample.