The Isomorphism Conjecture in algebraic K-theory

Let $\Gamma$ be a group. The group $K_0(\mathbb C\Gamma)$ is the Grothendieck group of finitely generated projective $\mathbb C\Gamma$-modules, and $\mathcal F in \Gamma$ denotes the family of finite subgroups of $\Gamma$. The Isomorphism Conjecture in algebraic $K$-theory predicts that $K_0(\mathbb C\Gamma)$ should be computable from the $K$-theory of the complex group rings of finite subgroups of $\Gamma$.

Isomorphism Conjecture in algebraic $K$-theory. The map from the colimit over finite subgroups

$$\operatorname{colim}_{H\in\mathcal F in \Gamma}K_0(\mathbb C H)\longrightarrow K_0(\mathbb C\Gamma)$$

should be an isomorphism. In particular, for a torsionfree group $\Gamma$,

$$K_0(\mathbb C\Gamma)\cong\mathbb Z.$$

This is a special case of the algebraic $K$-theory isomorphism conjecture discussed in the paper. The source gives no resolution status for this formulation.