Building on previous work by Davis and Lück, and recent constructions of bivariant K-theory as a stable \(\infty\)-category, I will sketch the construction of a Chern character running from groupoid-equivariant K-homology to (periodicized) homology groups of the given groupoid. Under the assumption that the isotropy groupoid is torsion free, this map is a rational isomorphism, thereby establishing a modified form of Matui’s HK conjecture (after integral counterexamples have been found). This construction also computes the rational homotopy type of the algebraic K-theory spectrum of ample groupoids as defined in a recent work by X. Li. This is joint work-in-progress with M. Yamashita.