The fully relatively quasi-convex virtual-retract conjecture

Let $N$ be an orientable, compact, non-positively curved $3$-manifold with empty or toroidal boundary, and write $\pi=\pi_1(N)$. Equip $\pi$ with its natural relatively hyperbolic structure. A subgroup is fully relatively quasi-convex when it is relatively quasi-convex and has, for each peripheral subgroup, either trivial intersection or finite-index intersection. Fully relatively quasi-convex virtual-retract conjecture. If $\Gamma\leq\pi$ is fully relatively quasi-convex, then $\Gamma$ is a virtual retract of $\pi$; in particular, $\Gamma$ is separable. The source presents this as a proposed class of separable subgroups and gives no resolution status.