The Whitehead group conjecture for torsionfree groups

Let $G$ be a group and let $\operatorname{Wh}(G)$ denote its Whitehead group. The Whitehead group conjecture. If $G$ is torsionfree, then $\operatorname{Wh}(G)$ vanishes. This would imply that every homotopy equivalence between finite $CW$-complexes with fundamental group $G$ is simple; the source gives no resolution status.