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Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?
In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...