Real-rank-zero conjecture for amenable group $C^*$-algebras

Let $G$ be a discrete, amenable group. Suppose that the full group $C^*$-algebra $C^*(G)$ has real rank zero.

Real-rank-zero conjecture. Then $G$ is locally finite.

This strengthens Effros's conjecture for amenable groups by replacing the AF-algebra assumption with the weaker property of real rank zero. The conjecture is stated as open in the source, although it has been verified for several classes of groups.