Gorenstein conjecture for tautological rings of moduli spaces of pointed curves

Let $\overline{M}_{g,n}$ be the moduli space of stable pointed curves, and consider the filtration

$$\overline{M}_{g,n} \supset M^c_{g,n} \supset M^{rt}_{g,n} \supset C_{g,n}.$$

Here $M^c_{g,n}$ is the locus of curves of compact type, $M^{rt}_{g,n}$ is the locus of curves with rational tails, and $C_{g,n}$ is the locus with a fixed stabilized complex structure. Let $R^*$ denote the tautological ring of each member of this filtration, obtained as the image of $R^*(\overline{M}_{g,n})$ in the associated quotient sequence. A finite-dimensional graded algebra $R$ is Gorenstein with socle in degree $s$ if there is an evaluation isomorphism $R^s\xrightarrow{\sim}\mathbb{Q}$ such that the product pairings $R^r\times R^{s-r}\to\mathbb{Q}$ are nondegenerate.

Gorenstein conjecture. The tautological rings of the filtration of $\overline{M}_{g,n}$ are finite-dimensional Gorenstein algebras.

This predicts that these tautological rings resemble the cohomology rings of compact manifolds, which are Gorenstein algebras. The conjecture concerns the structure of tautological rings across the boundary, compact-type, rational-tail, and fixed-curve strata; no resolution is specified here.