Virtually cocompactly cubulated Artin-Tits groups

We give a conjectural classification of virtually cocompactly cubulated Artin-Tits groups (i.e. having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin-Tits groups of spherical type, FC type or two-dimensional type. A particular case is that for $n geq 4$, the $n$-strand braid group is not virtually cocompactly cubulated.