Faber's tautological-algebra conjecture for the moduli space of curves

Let ${\mathbf M}_g$ be the moduli space of smooth genus-$g$ curves, and let ${\cal R}^*({\mathbf M}_g)$ be its tautological algebra, with tautological classes $\kappa_i$. Faber's conjecture. The algebra ${\cal R}^*({\mathbf M}_g)$ behaves like the cohomology algebra of a nonsingular projective variety of dimension $g-2$: it vanishes in degrees greater than $g-2$, is one dimensional in degree $g-2$, and the pairing

$${\cal R}^{i}({\mathbf M}_g)\times {\cal R}^{g-2-i}({\mathbf M}_g)\longrightarrow {\cal R}^{g-2}({\mathbf M}_g)$$

is perfect; it also satisfies Hard Lefschetz and Hodge Positivity with respect to $\kappa_1$. Moreover, the $[g/3]$ classes $\kappa_1,\ldots,\kappa_{[g/3]}$ generate the algebra with no relations in degrees at most $[g/3]$, and there are explicit formulas for the proportionalities in degree $g-2$. The source notes that the generation assertion is proved there at the level of rational cohomology, while the remaining structural and proportionality assertions are presented as conjectural.