Roth–Vakil affine stratification conjecture for partial moduli spaces of curves

Let $g,n>0$ satisfy $2g-2+n>0$. Let $\mathcal{M}^{\leq k}_{g,n}$ denote the moduli space of stable pointed curves of genus $g$ with at most $k$ rational components, and let $\mathcal{M}^{rt}_{g,n}$ and $\mathcal{M}^{ct}_{g,n}$ denote the corresponding moduli spaces with rational tails and of compact type, respectively. Their coarse spaces are considered with affine stratifications.

Roth–Vakil's conjecture. The coarse space of $\mathcal{M}^{\leq k}_{g,n}$ has an affine stratification with $g+k$ strata; the coarse spaces of $\mathcal{M}^{rt}_{g,n}$ and $\mathcal{M}^{ct}_{g,n}$ have affine stratifications with, respectively, $g+n-1$ and $2g-2+n$ strata.

These proposed stratifications would extend Looijenga's conjecture to important partial compactifications of the moduli space of pointed curves. They are motivated by vanishing results for tautological classes and would imply bounds on the cohomological and homotopical dimensions of these moduli spaces. Their status is not resolved in the supplied source.