Farrell–Jones conjecture for $K_0$ of torsion-free group rings

Let $G$ be a torsion-free group and let $R$ be a regular ring. Farrell–Jones conjecture for $K_0$. The map induced by the inclusion $R\to RG$ is an isomorphism

$$K_0(R)\longrightarrow K_0(RG).$$

This is described as a central open question in the $K_0$ theory of group rings; it predicts that the projective class group of $RG$ is detected by the coefficient ring for torsion-free groups.