Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?

Question

Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-dimensional compact, negatively curved, manifold? When $n=2$, such groups are known not to exist, and when $n=3$ this is an open question, originally posed by J. Cannon.

For $n\geq 4$ this MathOverflow answer gives a number of examples where $\Gamma$ is not the fundamental group of a hyperbolic manifold. However, these examples still arise as fundamental groups of negatively curved (albeit not hyperbolic) manifolds of the appropriate directions. Are there any known examples of hyperbolic groups with sphereical boundary which do not arise at all as the fundamental group of negatively curved compact manifolds?

Answer 1

The requirement that the manifolds "do not arise at all from negatively curved compact manifolds" is somewhat vague, but here is one known construction.

If one applies Charney-Davis strict hyperbolization to a closed oriented PL manifold with a non-integral Pontryagin number, the result is an aspherical manifold with hyperbolic fundamental group whose boundary is a sphere (by work of Davis and Januszkiewicz), and which is not the fundamental group of a closed aspherical smooth manifold. Such manifolds exist in all dimensions $4k-1$ where $k\ge 2$ is an integer. This is explained in "Aspherical manifolds with hyperbolic fundamental group" by Barthel, Lueck and Weinberger, on the bottom of p.12 .