Separability conjecture for relatively quasi-convex subgroups of relatively hyperbolic groups

Let $G$ be a group hyperbolic relative to the peripheral system $\mathcal{P}=\{P_1,\ldots,P_n\}$. Suppose that hyperbolic groups are residually finite, and that each $P_i$ is finitely generated and virtually nilpotent. A subgroup of $G$ is relatively quasi-convex if it is relatively quasi-convex with respect to this relatively hyperbolic structure. Separability conjecture. Every relatively quasi-convex subgroup of $G$ is separable. This would extend the paper's separability results from hyperbolic groups to relatively hyperbolic groups with finitely generated virtually nilpotent peripheral subgroups, assuming residual finiteness of hyperbolic groups; the claim is presented as a natural generalization and its resolution is not given here.