Bass's conjecture for commutative integral domains

Let $R$ be a commutative integral domain and $G$ a group. For $g\in G$, write $\operatorname{HS}_{RG}(P)(g)$ for the value at the conjugacy class $(g)$ of the Hattori–Stallings rank of a finitely generated projective $RG$-module $P$. Bass's conjecture. If $|g|$ is infinite, or if $|g|$ is finite and not invertible in $R$, then

$$\operatorname{HS}_{RG}(P)(g)=0.$$

This conjecture concerns the support of Hattori–Stallings ranks and has topological consequences, including statements about homotopy idempotents and Euler characteristics. It is known in several cases, such as certain linear groups and groups of bounded cohomological dimension, but remains open in general.