Stable subgroups of graph products

We extend the characterization of stable subgroups of right-angled Artin groups of Koberda, Mangahas and Taylor to the case of graph products of infinite groups. Specifically, we show that the stable subgroups of such graph products are exactly the subgroups that quasi-isometrically embed in the associated contact graph. Equivalently, they are the subgroups that satisfy a condition arising from the defining graph: a stable subgroup is an almost join-free subgroup. In particular, we generalize the equivalence between stable and purely loxodromic subgroups from Koberda, Mangahas and Taylor in the case where all torsion subgroups of the vertex groups are finite, and the equivalence between stable and infinite index Morse subgroups from Tran in the case where the defining graph is connected.