Faber's Gorenstein conjectures for the rational tails locus

Let $M_{g,n}^{rt}$ be the moduli space of stable curves with rational tails, and let $R^*(M_{g,n}^{rt})$ be its tautological ring. Let $R^*_c(M_{g,n}^{rt})$ denote the compactly supported tautological algebra, consisting of the elements of $R^*(\overline{M}_{g,n})$ that restrict trivially to $\overline{M}_{g,n} \backslash M_{g,n}^{rt}$. The intersection product makes $R^*_c(M_{g,n}^{rt})$ an $R^*(M_{g,n}^{rt})$-module.

Faber's Gorenstein conjectures. (D) The intersection pairings

$$R^k(M_{g,n}^{rt}) \times R^{3g-3+n-k}_c(M_{g,n}^{rt}) \rightarrow R_c^{3g-3+n}(M_{g,n}^{rt}) \cong \mathbb{Q}$$

are perfect for $k \geq 0$. (E) In addition to (D), $R^*_c(M_{g,n}^{rt})$ is a free $R^*(M_{g,n}^{rt})$-module of rank one.

These assertions generalize the compactly supported tautological-ring conjectures of Hain and Looijenga from $M_g$ to the rational tails locus. The source presents them as conjectures, and no resolution status is supplied here.