Faber's Gorenstein quotient conjecture from relations on the universal curve

Let $C_g^{2g-1}$ be the $(2g-1)$-fold fiber product of the universal curve over $M_g$, let $\mathbb{E}$ be the Hodge bundle, and let $K_i$ and $D_{ij}$ denote the tautological classes on $C_g^{2g-1}$. Let $\pi:C_g^{2g-1}\to M_g$ be the forgetful map. In the polynomial ring $\mathbb{Q}[\kappa_1,\ldots,\kappa_{g-2}]$, define $I_g$ to be generated by the classes

$$\pi_*(M\cdot c_j(\mathbb{F}_{2g-1}-\mathbb{E})),$$

where $j\geq g$ and $M$ is a monomial in the $K_i$ and $D_{ij}$. Faber's quotient conjecture. The quotient ring $\mathbb{Q}[\kappa_1,\ldots,\kappa_{g-2}]/I_g$ is Gorenstein with socle in degree $g-2$, and hence is isomorphic to $R^*(M_g)$. The construction seeks to recover the tautological ring from geometric relations; it is known to work in a range of genera, but the asserted Gorenstein property and identification are not known in general.