  We generalize a result of Paulin on the Gromov boundary of hyperbolic groups
to the Morse boundary of proper, maximal 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 2-stable,
quasi-m\"obius homeomorphism between their Morse boundaries. 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.
