Looijenga's affine covering conjecture for the moduli space of curves

Let $\mathcal{M}_g$ be the coarse moduli space of genus $g$ Riemann surfaces, viewed as a quasiprojective complex variety. Looijenga's conjecture. For every $g\geq 2$, the variety $\mathcal{M}_g$ can be covered by $(g-1)$ open affine subsets. This conjecture concerns the affine geometry of the moduli space and is used to predict the sharp coherent cohomological dimension; the source does not state a resolution in general. For $g=2$, the assertion says that $\mathcal{M}_2$ itself is affine, which follows because every genus $2$ Riemann surface is hyperelliptic.