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 $\mathrm{\Gamma }<G$ with respect to a parabolic subgroup $P_{\theta }$, we prove that any $\mathrm{\Gamma }$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G\mathrm{/}{P}_{\theta }$. More generally, we prove that for a Zariski dense $\theta$-transverse subgroup $\mathrm{\Gamma }<G$, any $(\mathrm{\Gamma },\psi )$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G\mathrm{/}{P}_{\theta }$, provided the $\psi$-Poincaré series of $\mathrm{\Gamma }$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups.