The Kadison–Kaplansky conjecture for torsion-free groups

Let $\Gamma$ be a torsion-free group, and let $C^*_r\Gamma$ be its reduced $C^*$-algebra. A projection in $C^*_r\Gamma$ is non-trivial if it is different from $0$ and $1$.

Kadison–Kaplansky conjecture. The algebra $C^*_r\Gamma$ contains no non-trivial projections. The conjecture is presented as a consequence of surjectivity of the Baum–Connes assembly map and the trace conjecture. The source does not state whether it has been resolved.