Faber's strong conjecture for the compactly supported tautological algebra

Let $\mathcal R^\bullet({\mathcal M}_g)$ be the tautological algebra of the moduli space of smooth genus-$g$ curves, and let $\mathcal R_c^\bullet({\mathcal M}_g)$ be its compactly supported tautological ideal. The intersection product gives pairings

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

Faber's strong conjecture. These pairings are perfect for $k=0,1,2,\dots$, and $\mathcal R_c^\bullet({\mathcal M}_g)$ is a free $\mathcal R^\bullet({\mathcal M}_g)$-module of rank one. The source reports evidence, including verification for genera at most $15$, but presents the conjecture as unresolved in general.