  Let X be a proper CAT(0) cube complex admitting a proper cocompact action by
a group G. We give three conditions on the action, any one of which ensures
that X has a factor system in the sense of [BHS14]. We also prove that one of
these conditions is necessary. This combines with results of
Behrstock--Hagen--Sisto to show that $G$ is a hierarchically hyperbolic group;
this partially answers questions raised by those authors. Under any of these
conditions, our results also affirm a conjecture of BehrstockHagen on
boundaries of cube complexes, which implies that X cannot contain a convex
staircase. The conditions on the action are all strictly weaker than virtual
cospecialness, and we are not aware of a cocompactly cubulated group that does
not satisfy at least one of the conditions.
