Kapovich–Kleiner conjecture for Sierpiński carpet boundaries

Let $G$ be a hyperbolic group, and let $S$ be the Sierpiński carpet. Let $\partial G$ denote the boundary of $G$, and let $\mathbb{H}^3$ denote hyperbolic 3-space. Kapovich–Kleiner conjecture. If $\partial G\cong S$, then $G$ acts geometrically on a convex subset of $\mathbb{H}^3$ bounded by totally geodesic planes. This is a rigidity question for hyperbolic groups with Sierpiński carpet boundary, predicting that such groups arise geometrically from convex co-compact Kleinian actions. Its status is not resolved in the supplied source.