The reduced projective class group conjecture for torsionfree groups

Let $G$ be a group and let $\widetilde{K}_0(\mathbb{Z}G)$ denote its reduced projective class group. The reduced projective class group conjecture. The group $\widetilde{K}_0(\mathbb{Z}G)$ vanishes if $G$ is torsionfree. This is an algebraic form of the finiteness-obstruction problem for finitely dominated $CW$-complexes; the source gives no resolution status.