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 its cohomology is a Koszul algebra.
Question: I was wondering if there are any similar results for $\overline{\mathcal{M}}_{g,n}$. These spaces are known to be formal (at least over the rationals, see https://arxiv.org/abs/math/0402098) , so what about coformality?
An easiest question: do we know if $\overline{\mathcal{M}}_{g,n}$ is coformal at least for small values of $g\geq 1$ and $n\geq 0$? For example $H^*(\overline{\mathcal{M}}_{1,4};\mathbb{Q})$ should be known as a ring (I think the answer is more or less contained in this paper of Getzler https://arxiv.org/abs/alg-geom/9612004), so one can try to see if the cohomology is a Koszul algebra. Do anyone knows the answer?