Novikov's conjecture for algebraic K-theory

Let $R$ be one of $\mathbb Z$, $\mathbb Q$, $\mathbb R$, or $\mathbb C$, and let $\pi$ be a discrete group. Let $B\pi$ be a classifying space and let $\mathbb K(R)$ and $\mathbb K(R\pi)$ denote the relevant algebraic $K$-theory spectra.

Novikov conjecture for K-theory. The assembly map

$$B\pi_+\wedge \mathbb K(R)\to \mathbb K(R\pi)$$

induces an injection on rational homotopy groups.

This is the $K$-theoretic analogue of Novikov's conjecture. The source records proofs in several important classes of groups, but the statement is not known for all discrete groups.