Divergent torus orbits in homogeneous spaces of Q-rank two

Let G be the real points of a semisimple algebraic Q-group, let H be an arithmetic subgroup of G and let T be the real points of an R-split torus in G. We prove that if there is a divergent T-orbit in G/H, and Q-rank(G) > 1, then the dimension of T is not larger than Q-rank(G). This provides a partial answer to a question of G.Tomanov and B.Weiss.