The relative Cannon conjecture

Let $(G,\mathcal{P})$ be a relatively hyperbolic group pair with $G$ torsion-free. Its Bowditch boundary is denoted by $\pa_B(G,\mathcal{P})$, and $M$ denotes a finite-volume hyperbolic $3$-manifold. The peripheral groups are the groups in $\mathcal{P}$.

The relative Cannon conjecture. If

$$\pa_B(G,\mathcal{P})\simeq S^2,$$

then $G$ is the fundamental group of $M$. Furthermore, the peripheral groups are the fundamental groups of the cusps and totally geodesic boundary components of $M$.

This is a relative form of the Cannon conjecture. The paper proves that the Wall conjecture implies this conjecture, but the source does not establish either conjecture itself.