Rational K(pi,1) conjecture for the real moduli space

Let $M_n$ be the real locus of the moduli space of stable genus-zero curves with $n$ marked points. A connected space is rational $K(\pi,1)$ if its rational completion is a $K(\pi,1)$-space. Rational K(pi,1) conjecture. The space $M_n$ is rational $K(\pi,1).$ This follows from the Koszulness conjecture if the cited criterion applies, but remains open in general.