No-wandering-continua conjecture for cocompact Kleinian groups

Let $G$ be a cocompact Kleinian group acting on the sphere at infinity $S^2_\infty$, and let $K\subset S^2_\infty$ be a compact connected set disjoint from every $G$-translate of itself. No-wandering-continua conjecture. Then $K$ consists of a single point. The claim rules out nontrivial compact connected subsets of the sphere at infinity that are disjoint from all their translates.