R_-property for groups commensurable to nilpotent quotients of RAAGs

Let G be a group and an automorphism of G. Two elements x, y G are said to be -conjugate if there exists a third element z G such that z x (z) ^-1 = y. Being -conjugate defines an equivalence relation on G. The group G is said to have the R_-property if all its automorphisms have infinitely many -conjugacy classes. For finitely generated torsion-free nilpotent groups, the so-called Mal'cev completion of the group is a useful tool in studying this property. Two groups have isomorphic Mal'cev completions if and only if they are abstractly commensurable. This raises the question whether the R_-property is invariant under abstract commensurability within the class of finitely generated torsion-free nilpotent groups. We show that the answer to this question is negative and provide counterexamples within a class of 2-step nilpotent groups associated to edge-weighted graphs. These groups are commensurable to 2-step nilpotent quotients of right-angled Artin groups.