On Koszulity in homology of moduli spaces of stable n-pointed curves of genus zero

We prove Koszulity of the homology of the of moduli spaces of stable n-pointed curves of genus zero $\overline{M}_{0,n}$ for $n=5$, using its presentation due to Keel and the Priddy criterion of Koszulity. For $n=6 $ we establish that $\overline{M}_{0,6}^!$ is potential, find the expression for the potential, and based on that prove that it is Koszul.