The Gorenstein property and Pixton's conjecture for compact type moduli

We show that the tautological ring of $\mathcal{M}_{g,n}^{\mathrm{ct}}$ is not Gorenstein for $g\geq 2$ and $2g+n\geq 12$. We prove new cases of Pixton's conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including $\mathcal{M}_{6}^{\mathrm{ct}}$, $\mathcal{M}_{5,2}^{\mathrm{ct}}$, and $\mathcal{M}_7^{\mathrm{ct}}$. These are the first known cases where Pixton's conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.