Integrality of the canonical trace

Prove that for every torsion-free discrete group G, the homomorphism induced by the canonical trace, $(\tau_G)_*:K_0(C_r^*(G))\to\mathbb{R}$, has image exactly equal to $\mathbb{Z}$.

Background

The canonical trace on C_r^*(G) induces a homomorphism on K_0. The conjecture asserts that its range is integral for torsion-free groups. The paper shows that this statement would imply the Kaplansky–Kadison conjecture and that the Baum–Connes conjecture implies it via surjectivity of the assembly map.