The taut 20 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 let the tautological subspace be the subspace generated by tautological classes.

Taut 20 conjecture. The cohomology groups

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

are tautological for all even $k\leq20$, and for all $g$ and $n$.

The conjecture extends known tautological-generation results and is attributed in the source to Canning--Larson--Payne. Its bound is sharp because degree-22 cohomology is non-tautological in examples such as $\overline{\mathcal{M}}_{2,20}$ and $\overline{\mathcal{M}}_{12}$; it remains open.