Toral relative Cannon conjecture

Let $(G,P)$ be a relatively hyperbolic group pair, meaning that $G$ is relatively hyperbolic relative to the collection of subgroups $P$. Assume that its Bowditch boundary is homeomorphic to $S^2$, that $P\neq\emptyset$, and that every element of $P$ is $\mathbb{Z}^2$. Toral relative Cannon conjecture. Then $G$ is Kleinian. This is a relative analogue of the Cannon conjecture motivated by the greater tractability of non-uniform lattices, whose peripheral subgroups are $\mathbb{Z}^2$ and whose Bowditch boundary is a $2$-sphere. The paper presents this as a relative version that may be more approachable, while the corresponding conjecture remains unresolved.