Hain–Looijenga perfect-pairing and freeness conjecture for compactly supported tautological classes

Let $M_g$ be the moduli space of smooth curves of genus $g$, and let $R_c^*(M_g)$ be the ideal of classes in $R^*(\overline{M}_g)$ restricting trivially to the Deligne–Mumford boundary. The intersection product gives pairings

$$R^k(M_g)\times R_c^{3g-3-k}(M_g)\longrightarrow R_c^{3g-3}(M_g)\cong\mathbb{Q}.$$

Hain–Looijenga conjecture. These pairings are perfect for every $k\geq0$; in addition, $R_c^*(M_g)$ is a free $R^*(M_g)$-module of rank one. This conjecture proposes a form of duality for compactly supported tautological classes and remains open in general.