The revised rational HK-conjecture for étale groupoids

Let $\mathcal{G}$ be a second countable, locally compact, Hausdorff, étale groupoid, and let $\widehat{\mathcal{G}}$ be its blow-up at torsion elements of isotropy groups. Write $H_{**}(\widehat{\mathcal{G}};\mathbb{C}_{\widehat{\mathcal{G}}^{(0)}})$ for the associated $\mathbb{Z}/2$-graded Crainic–Moerdijk homology, and suppose that $c$-$\mathbb{C}$-$\operatorname{dim}(\widehat{\mathcal{G}}^{(0)})<\infty$. The revised rational HK-conjecture. If the rational Baum–Connes conjecture holds for $\mathcal{G}$, then

$$K_*(C^*_r(\mathcal{G}))\otimes \mathbb{C}\cong H_{**}(\widehat{\mathcal{G}};\mathbb{C}_{\widehat{\mathcal{G}}^{(0)}}).$$

This revises Matui’s HK-conjecture in the presence of torsion isotropy and also applies beyond groupoids with zero-dimensional unit space. The paper proves it for a large class of transformation groupoids, while the general assertion remains open.