The comparison conjecture for Schatten-class and compact-operator K-theory

Let $\Gamma$ be a group. Let $K$ be the $C^*$-algebra of all compact operators on an infinite-dimensional separable Hilbert space, and let $C_r^*(\Gamma)$ be the reduced group $C^*$-algebra of $\Gamma$. Comparison conjecture. The natural homomorphism

$$i_*:K_n({\cal S}\Gamma)\longrightarrow K_n(C_r^*(\Gamma)\otimes K)$$

is an isomorphism, where $C_r^*(\Gamma)\otimes K$ is the $C^*$-algebraic tensor product and $i$ is the inclusion of ${\cal S}\Gamma$ into $C_r^*(\Gamma)\otimes K$. The source presents this as a consequence expected from the Farrell–Jones and Baum–Connes isomorphism conjectures; the supplied text does not establish it in general.