Xiao–Reid's moduli-stack Morsification conjecture

Let $g$ be the genus, and let $\operatorname{M}_g$ be the moduli stack of smooth curves of genus $g$. A moduli stack of good smoothable curves is a stack whose geometric points include all good smoothable curves of genus $g$.

Xiao–Reid's Morsification conjecture. There exists a moduli stack $\mathcal{M}^{\star}_{g}$ of curves which contains all good smoothable curves of genus $g$ as geometric points such that any divisorial component of the complement of $\mathcal{M}_{g}$ in $\mathcal{M}^{\star}_{g}$ generically parametrizes stable curves with one node or smooth multiple curves.

This conjecture is proposed as a way to extend the paper's arguments to the non-reduced case by controlling the divisorial boundary of a moduli stack containing good smoothable curves. Its resolution is presented as necessary for that extension.