The rational Novikov conjecture for assembly maps

Let $\pi$ be a group, and let $A^L_\pi$ and $A^K_\pi$ denote the assembly maps in $L$-theory and $K$-theory, respectively. Their rationalizations are obtained by tensoring with the identity of $\mathbb Q$. The rational Novikov conjecture. For any group $\pi$, the rational assembly maps $A^L_\pi\otimes\mathrm{id}_{\mathbb Q}$ and $A^K_\pi\otimes\mathrm{id}_{\mathbb Q}$ are injective. This is one of the outstanding assembly-map conjectures in algebraic surgery and algebraic $K$-theory; it remains open in the generality stated, even for groups whose classifying spaces can be realized as closed aspherical manifolds.