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

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring $R^{\bullet}(\overline {\mathcal {M}}_{2,N})$ is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on $\overline {\mathcal {M}}_{2,N}$ . By work of Graber and Pandharipande, such a class exists on $\overline {\mathcal {M}}_{2,20}$ , and we present some evidence indicating that N is in fact 20.