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Let $N$ be the smallest integer such that there is a non-tautologicalcohomology class of even degree on $\mathcal{\bar M}_{2,N}$. We remark thatthere is such a non-tautological class on $\mathcal{\bar M}_{2,20}$, by work ofGraber and Pandharipande. We show that $\mathcal{\bar M}_{2,N}$ hasnon-tautological cohomology only in one degree, which is not the middle degree.In particular, it follows that the tautological ring of $\mathcal{\barM}_{2,N}$ is not Gorenstein. We present some evidence suggesting that N=20holds.

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\bar M}_{2,n}$