Gromov's conjecture on hyperbolic groups with spherical boundary

Let $G$ be a hyperbolic group whose boundary $\partial G$ is homeomorphic to $S^{n-1}$. An aspherical closed manifold is a closed manifold whose universal cover is contractible. Gromov's conjecture. $G$ is the fundamental group of an aspherical closed manifold of dimension $n$. This conjecture is motivated by the fact that the fundamental group of a closed negatively curved $n$-manifold is hyperbolic with boundary homeomorphic to $S^{n-1}$. The general realization problem remains open.