Horospherical invariant measures and a rank dichotomy for Anosov groups

Let $ G = \prod_{i 1}^{\mathsf r} G_i be a product of simple real algebraic groups rank one and \Gamma an Anosov subgroup with respect to minimal parabolic subgroup. For each \mathsf v in the interior positive Weyl chamber, let \mathscr R_ v\subset \Gamma\backslash denote Borel subset all points recurrent \exp (\mathbb R_+ v) $-orbits. maximal horospherical N $, we show that $-action on {\mathscr R}_ is uniquely ergodic if r \operatorname{rank}(G)\le 3 belongs limit cone there exists no $-invariant Radon measure otherwise.