Relative biexactness conjecture for relatively hyperbolic groups

Let $G$ be a group, and let $\{H_i\}_{i=1}^{n}$ be a family of peripheral subgroups. Assume that $G$ is exact and hyperbolic relative to $\{H_i\}_{i=1}^{n}$.

Relative biexactness conjecture. $G$ is biexact relative to $\{H_i\}_{i=1}^{n}$.

This folklore conjecture would provide another family of examples to which the paper's results on relatively biexact group von Neumann algebras apply. The source does not state a resolution, so its status remains open.