The general HK-conjecture for étale groupoids

Let $\mathcal{G}$ be a second countable, étale groupoid, and let $\widehat{\mathcal{G}}$ be the blow-up groupoid with base-related notation $\widehat{X}$. Suppose that $c$-$\mathbb{C}$-$\operatorname{dim}(\widehat{X})<\infty$, and that the Baum–Connes conjecture holds for $\mathcal{G}$. The general HK-conjecture. Then

$$K_*(C^*_r(\mathcal{G}))\otimes \mathbb{C}\cong H_{**}(\widehat{\mathcal{G}};\mathbb{C}_{\widehat{X}}).$$

This is the unrevised general formulation preceding the rational version. The paper establishes the corresponding result for transformation groupoids, while the general conjecture is not stated to be proved.