Hierarchical hyperbolicity conjecture for parabolically geometrically finite subgroups

Let $S$ be a surface and let $G<\operatorname{Mod}(S)$ be parabolically geometrically finite, meaning that $G$ is relatively hyperbolic relative to a possibly trivial collection of subgroups $\mathcal H=\{H_1,\ldots,H_k\}$, each $H_i$ contains a finite-index abelian subgroup consisting entirely of multitwists, and the coned-off Cayley graph of $G$ admits a $G$-equivariant quasi-isometric embedding into the curve graph $\mathcal C(S)$. Let $\Gamma$ be the $\pi_1S$-extension group of $G$. Hierarchical hyperbolicity conjecture. The group $\Gamma$ is a hierarchically hyperbolic group. This proposes that the hierarchical hyperbolicity established for the relevant extension groups under the paper's preceding hypotheses should extend to all parabolically geometrically finite subgroups; the source provides no resolution of this proposed statement.