  We use the projection complex machinery of Bestvina--Bromberg--Fujiwara to
study hierarchically hyperbolic groups. In particular, we show that if the
group has a BBF colouring and its associated hyperbolic spaces are
quasiisometric to trees, then the group is quasiisometric to a
finite-dimensional CAT(0) cube complex. We deduce various properties, including
the Helly property for hierarchically quasiconvex subsets.
