The atoroidal Poincaré duality group conjecture

Let $G$ be a $3$-dimensional Poincaré duality group. Call $G$ atoroidal if it contains no subgroup isomorphic to $\mathbb Z\oplus\mathbb Z$. Atoroidal hyperbolicity conjecture. Every atoroidal $3$-dimensional Poincaré duality group is hyperbolic. This would identify the atoroidal case in the proposed three-way decomposition of $3$-dimensional Poincaré duality groups and is related to the Cannon conjecture.