Hierarchically hyperbolic groups are determined by their Morse boundaries

We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of top-heavy hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a homeomorphism between their Morse boundaries such that the map and its inverse are 2-stable, quasi-mobius. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental groups of 3-manifolds without Nil or Sol components.