Property QT of relatively hierarchically hyperbolic groups

Using the projection complex machinery, Bestvina--Bromberg--Fujiwara, Hagen--Petyt, and Han--Nguyen--Yang have proved that several classes of nonpositively curved groups admit equivariant quasi-isometric embeddings into finite products of quasi-trees, i.e. having property QT. In this paper, we unify and generalize these results by establishing a sufficient condition for relatively hierarchically hyperbolic groups to have property QT. As applications, we show that a group has property QT if it is residually finite and belongs to one of the following classes of groups: admissible groups, hyperbolic-$2$-decomposable groups with no distorted elements, Artin groups of large and hyperbolic type, and $π_1$-extension groups of lattice Veech groups. We also introduce a slightly stronger version of property QT, called property QT$_0$, and show the invariance of property QT$_0$ under graph products.