Friedlander–Milnor conjecture for Lie groups

Let $G$ be a Lie group. Regard $BG$ as the classifying space of the underlying discrete topological group, and let $BG^{\rm top}$ be the classifying space with its usual topology. The canonical map

$$\psi:BG\longrightarrow BG^{\rm top}$$

is induced by the identity on the underlying group. Friedlander–Milnor conjecture. The map $\psi$ induces isomorphisms on homology and cohomology with every finite abelian coefficient group. This is a comparison conjecture between discrete and topological classifying spaces of Lie groups.