The Gorenstein conjecture fails for the tautological ring of M₂,ₙ

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Let N be the smallest integer such that there is a non-tautological cohomology class of even degree on M₂,ₙ. We remark that there is such a non-tautological class on M₂,₂₀, by work of Graber and Pandharipande. We show that M₂,ₙ has non-tautological cohomology only in one degree, which is not the middle degree. In particular, it follows that the tautological ring of M₂,ₙ is not Gorenstein. We present some evidence suggesting that N=20 holds.