  We show that cubulated hyperbolic groups with spherical boundary of dimension
3 or at least 5 are virtually fundamental groups of closed, orientable,
aspherical manifolds, provided that there are sufficiently many quasi-convex,
codimension-1 subgroups whose limit sets are locally flat subspheres. The proof
is based on ideas used by Markovic in his work on Cannon's conjecture for
cubulated hyperbolic groups with 2-sphere boundary.
