Hsiang's finiteness-obstruction conjecture for finitely dominated classifying spaces

Let $\Gamma$ be a group and let $B\Gamma$ denote its classifying space. Assume that $B\Gamma$ is finitely dominated, and denote its finiteness obstruction by $\sigma(B\Gamma)$. Hsiang's finiteness-obstruction conjecture. One has

$$\sigma(B\Gamma)=0.$$

This conjecture has been verified in many cases, including cases where $\widetilde K_0(\mathbb Z[\Gamma])=0$. The source explains that it follows from Hsiang's assembly-map conjecture for torsion-free groups, which asserts that the assembly map is a homotopy equivalence of spectra.