The taut 10 conjecture for moduli spaces of stable curves

For the moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable curves, let $H^k(\overline{\mathcal{M}}_{g,n})$ denote rational singular cohomology, and call it tautological when it is generated by tautological classes.

Taut 10 conjecture. All cohomology groups

$$H^k(\overline{\mathcal{M}}_{g,n})$$

are tautological for $k\leq10$.

The bound is best possible because $H^{11}(\overline{\mathcal{M}}_{1,11})$ is not tautological. The conjecture follows from the taut 20 conjecture together with the vanishing of odd-degree cohomology below degree 11, but it remains open.