BGGH's conjecture on Cannon–Thurston maps for HHG pairs

Let $G$ be a hierarchically hyperbolic group (HHG), and let $H<G$ be an infinite hyperbolic normal subgroup of infinite index. A Cannon–Thurston map is a continuous extension of the inclusion $H\hookrightarrow G$ to the relevant hierarchical boundaries. BGGH's conjecture. The Cannon–Thurston map does not exist for $(H,G)$ unless either $G$ is hyperbolic or $H$ is hierarchically quasiconvex in some HHG structure on $G$. This conjecture concerns the existence of boundary extensions for hyperbolic normal subgroups of infinite index in HHGs; the supplied source does not state whether it has been resolved.