Looijenga's affine stratification conjecture for moduli spaces of pointed curves

Let $g,n\geq 0$ be such that $2g-2+n>0$. The coarse space $M_{g,n}$ of $\mathcal{M}_{g,n}$ is the moduli space of smooth pointed curves of genus $g$ with $n$ marked points.

$$M_{g,n}=\coprod_{i=1}^{g-\delta_{n,0}} S_i,\qquad \overline{S}_j=\coprod_{i\leq j}S_i\quad\text{for all }j.$$

Looijenga's conjecture. There is such a stratification in which every locally closed stratum $S_i$ is affine.

The conjecture seeks a geometric explanation for vanishing results for tautological classes on moduli spaces of pointed curves. If true, the affine stratification would also give bounds on cohomological and homotopical dimensions. Its status is not resolved in the supplied source.