Faber–Pandharipande Gorenstein conjecture for the rational-tail tautological ring

Let $M_{g,n}^{\mathrm{rt}}$ be the moduli space of stable curves with rational tails, and let $R^*(M_{g,n}^{\mathrm{rt}})$ denote its tautological ring. Let $\delta_{0g}$ equal $1$ when $g=0$ and $0$ otherwise.

Faber–Pandharipande's rational-tail Gorenstein conjecture. The ring $R^*(M_{g,n}^{\mathrm{rt}})$ is Gorenstein with socle in degree

$$g-2+n-\delta_{0g}.$$

This is one of the conjectures concerning Gorenstein properties of tautological rings; the source records it as a conjecture without giving a resolution.