Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabbai, the limit set of $\Gamma$ is either the entire sphere $S^2\cong \partial \mathbb H^3$, or has zero area. (This result had first been conjectured by Ahlfors, hence the name.)

Are there any known counterexamples to the corresponding result for higher dimensions? What if we add the tameness restriction that $\mathbb{H^n}/\Gamma$ has a manifold compactification?