The Whitehead vanishing conjecture for torsion-free groups

For a group $G$, the Whitehead group $\operatorname{Wh}(G)$ is the quotient of $K_1(\mathbb{Z}[G])$ by the subgroup represented by the units $\pm g$ for $g\in G$. The Whitehead vanishing conjecture. If $G$ is torsion-free, then

$$\operatorname{Wh}(G)=0.$$

This folklore conjecture is equivalent, through the $s$-Cobordism Theorem, to the trivial h-cobordism conjecture for manifolds with fundamental group $G$; the source records several early cases but leaves the general statement open.