Convergence-group conjecture for mapping torus subgroups

Let $G$ be a tame mixed pseudo-fibered group, and let $H$ be the maximal purely hyperbolic subgroup of $G$. A p-A-like element is an element of the type defined in the paper, and for such an element $\phi$ define the mapping torus subgroup $G_\phi=H\rtimes\langle\phi\rangle$, with monodromy $\phi$.

Mapping torus convergence-group conjecture. For any p-A-like element $\phi$ of $G$, the mapping torus subgroup $G_\phi$ acts on $S^2$ as a convergence group.

This conjecture extends the paper's theorem that purely hyperbolic pseudo-fibered groups act on $S^2$ as convergence groups. The proposed extension remains open.