The virtually Kleinian conjecture for relatively hyperbolic groups with planar boundary

Let $G$ be a one-ended group and let $(G,\mathcal{P})$ be a relatively hyperbolic pair, with Bowditch boundary $\partial(G,\mathcal{P})$. Assume that this boundary is planar and has no cut points. Virtually Kleinian conjecture. Then $G$ is virtually a Kleinian group. This conjecture extends the Cannon conjecture. It is known in the case where the boundary is a $2$-sphere, giving the Relative Cannon Conjecture, and is also known in a key hyperbolic case without Sierpiński carpets and with no $2$-torsion. The Sierpiński carpet case is attributed to Kapovich and Kleiner; examples show that the virtual qualifier is necessary in general.