Coarse-convergence conjecture for Morse boundaries of hierarchically hyperbolic groups

Let $G$ be a hierarchically hyperbolic group (HHG), and let $\partial_M G$ be its Morse boundary. Assume that $G$ admits a largest acylindrical action. The topology on $\partial_M G$ induced by this action is the topology of coarse convergence of hierarchy paths. Coarse-convergence conjecture. The topology on the Morse boundary of an HHG induced by its largest acylindrical action is the topology of coarse convergence of hierarchy paths. Largest acylindrical actions are known to exist for all hierarchically hyperbolic groups and to be $\kappa$-injective, but identifying the induced topology with coarse convergence of hierarchy paths remains open.