Non-concentration property of Patterson-Sullivan measures for Anosov subgroups

Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma<G$ with respect to a parabolic subgroup $P_\theta$, we prove that any $\Gamma$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_\theta$. More generally, we prove that for a Zariski dense $\theta$-transverse subgroup $\Gamma<G$, any $(\Gamma, \psi)$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_\theta$, provided the $\psi$-Poincar\'e series of $\Gamma$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups.