Classical Farrell–Jones conjecture for regular group rings

Let $k$ be a regular ring, let $G$ be a torsionfree group, let $\mathrm{B}G$ denote its classifying space, and let $\mathbf{K}(k)$ denote the algebraic $K$-theory spectrum of $k$. Classical Farrell–Jones conjecture. For every $n\in\mathds{Z}$, the assembly map

$$\mathrm{H}_{n}(\mathrm{B}G; \mathbf{K}(k))\longrightarrow \mathrm{K}_{n}(k[G])$$

is an isomorphism. The conjecture predicts that the algebraic $K$-theory of the group ring $k[G]$ is determined by the homology of the classifying space of $G$ with coefficients in the $K$-theory spectrum of $k$; the paper assumes this conjecture in deriving its main result, but the supplied text gives no resolution of it.