Planarity conjecture for relatively hyperbolic groups

Let $(G,\mathcal{P})$ be a non-elementary relatively hyperbolic group pair, and let $\partial_B(G,\mathcal{P})$ denote its Bowditch boundary. A topological space is planar if it embeds in $S^2$; the boundary has no cut points when it contains no cut point.

Planarity conjecture. If $\partial_B(G,\mathcal{P})$ is planar and has no cut points, then $G$ is virtually isomorphic to a Kleinian group.

This conjecture extends the hyperbolic-group planarity question to relatively hyperbolic pairs. The exclusion of cut points is necessary in the source's discussion, and examples with planar Bowditch boundary that are not virtually Kleinian are noted; the resolution of the conjecture itself is not specified.