Hain and Looijenga's compactly supported tautological duality conjecture

Let $R^{\bullet}(M_g)$ be the tautological algebra of the moduli space $M_g$ of smooth genus-$g$ curves, and let $R^{\bullet}_c(M_g)$ be the compactly supported tautological ideal consisting of classes in $R^{\bullet}(\overline{M}_g)$ whose pull-back to every boundary stratum vanishes. The intersection product gives pairings

$$R^k(M_g)\times R^{3g-3-k}_c(M_g)\to R^{3g-3}_c(M_g)\cong\mathbb{Q}, \qquad k=0,1,2,\dots$$

Hain and Looijenga's conjecture. These intersection pairings are perfect (Poincaré duality), and $R^{\bullet}_c(M_g)$ is a free $R^{\bullet}(M_g)$-module of rank one.

This is a compactly supported analogue of the Gorenstein and Poincaré-duality conjectures for tautological rings. The conjecture is formulated here in the case $n=0$; the supplied source does not state whether it has been resolved.