The strong Novikov conjecture

Let $B\pi$ be the classifying space of a discrete group $\pi$, and consider the assembly map

$$\beta:K_*(B\pi)\to K_*(C^*(\pi)).$$

Here $C^*(\pi)$ is the group $C^*$-algebra. Strong Novikov conjecture. The map $\beta$ is rationally injective. The conjecture strengthens the homotopy-invariance problem for higher signatures through operator-algebraic $K$-theory; the source states that Kasparov's proof applies under the same stated hypotheses as his theorem but gives no general resolution.