The regular-ring projective class group conjecture for torsionfree groups

Let $G$ be a group and let $R$ be a regular ring, meaning that $R$ is Noetherian and every $R$-module has a finite-dimensional projective resolution. The regular-ring projective class group conjecture. If $G$ is torsionfree, then the canonical map

$$K_0(R) \xrightarrow{\cong} K_0(RG)$$

is bijective. In particular, $\widetilde{K}_0(RG)$ vanishes if $G$ is torsionfree and $R$ is a principal ideal domain. This generalizes the preceding conjecture; the source gives no resolution status.