The totally nonnegative wonderful compactification of G is a ball

Let $\overline{G}$ denote the wonderful compactification of a semisimple algebraic group $G$ of adjoint type. We use contractive flows to show that the totally nonnegative part of every $G \times G$ orbit closure of $\overline{G}$ is homeomorphic to a closed Euclidean ball. In particular, the totally nonnegative part of $\overline{G}$ is homeomorphic to a closed ball.