Koszulness conjecture for the cohomology algebra of the real moduli space

Let $M_n$ denote the real locus of the moduli space of stable genus-zero curves with $n$ marked points, and let $\Lambda_n=H^*(M_n,\mathbb{Q})$. Let $U_n=\Lambda_n^!\otimes\mathbb{Q}$ be its quadratic dual, with Hilbert series $P_n^!(t)$. Koszulness conjecture. The algebra $\Lambda_n\otimes\mathbb{Q}$, equivalently $U_n\otimes\mathbb{Q}$, is Koszul. In particular,

$$P_n^!(t)=\frac{1}{P_n(-t)}=\prod_{0\le k<(n-3)/2}(1-(n-3-2k)^2t)^{-1}.$$

This is known for $n\leq 6$, and the Hilbert-series formula has been verified computationally in degree $3$ for $n\leq 9$; the general conjecture remains open.