Boundary coarse Baum–Connes conjecture

Let $X$ be a uniformly discrete bounded geometry metric space, and let $A_{\partial}=l^{\infty}(X,\mathcal{K})/C_{0}(X,\mathcal{K})$. Writing $G(X)|_{\partial\beta X}$ for the restriction of the coarse groupoid to the boundary of the Stone–Čech compactification, consider the assembly map

$$\mu_{bdry}:K_{*}^{top}(G(X)|_{\partial\beta X},A_{\partial})\longrightarrow K_{*}(A_{\partial}\rtimes_{r}G(X)|_{\partial\beta X}).$$

Boundary coarse Baum–Connes conjecture. This assembly map is an isomorphism. This conjecture concerns the boundary version of the coarse Baum–Connes conjecture and is used in applications to coarse geometry and the coarse Novikov conjecture; its resolution is not specified in the source.