Equivariant coarse Baum–Connes conjecture (bijectivity of the assembly map)

Prove that for every proper metric space X with bounded geometry and every countable discrete group Γ acting properly and isometrically on X, the equivariant coarse assembly map μ_X^Γ is (rationally) bijective, i.e., an isomorphism on K-theory (or after tensoring with ℚ).

Background

This conjecture strengthens the equivariant coarse Novikov conjecture by asking not only injectivity but full isomorphism between the topological and analytic sides. It generalizes the Baum–Connes philosophy to the coarse, equivariant framework.

The authors position their results relative to this stronger goal, noting that their main theorems establish rational injectivity under geometric hypotheses, while full bijectivity remains conjectural in general.