Intersections of diagonal orbits

Let $A\subseteq S{L}_n(\mathbb{R})$ group of diagonal matrices with positive diagonal, let ${\mathrm{S}\mathrm{T}}_n\subseteq X_n:=S{L}_n(\mathbb{R})\mathrm{/}S{L}_n(\mathbb{Z})$ be the set of stable lattices, and let ${\mathrm{W}\mathrm{R}}_n\subseteq X_n$ be the set of well-rounded lattices. We prove that any $A$-orbit in $X_n$ intersects both ${\mathrm{S}\mathrm{T}}_n$ and ${\mathrm{W}\mathrm{R}}_n$.