The relative bi-exactness conjecture for relatively hyperbolic groups

Let $G$ be a finitely generated group hyperbolic relative to a collection of subgroups $\{H_\mu\}_{\mu \in \Lambda}$ of $G$. Relative bi-exactness conjecture. If all subgroups $H_\mu$ are exact, then $G$ is bi-exact relative to $\{H_\mu\}_{\mu \in \Lambda}$. This folklore conjecture concerns whether exact peripheral subgroups suffice for relative bi-exactness. The paper proves the related assertion that $G$ is bi-exact if and only if all peripheral subgroups are bi-exact, but the relative statement above is presented as a conjecture.