  We prove that the hierarchical hulls of finite sets of points in mapping
class groups and Teichm\"uller spaces are stably approximated by a CAT(0) cube
complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we
prove that mapping class groups are semihyperbolic and Teichm\"uller spaces are
coarsely equivariantly bicombable, and both admit stable coarse barycenters.
Our results apply to the broader class of "colorable" hierarchically hyperbolic
spaces and groups.
