Even-degree tautological cohomology conjecture for moduli spaces of curves

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable curves, let $H^k(\overline{\mathcal{M}}_{g,n})$ denote its cohomology, and let $RH^k(\overline{\mathcal{M}}_{g,n})$ denote the corresponding tautological cohomology. Even-degree tautological cohomology conjecture. For every even $k\leq 20$ and all $g,n$, one has

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

The claim is motivated by the expectation that even cohomology in this range is pure Tate, while known non-tautological algebraic cycles occur in degrees at least $22$; the source does not report a resolution.