The bubble-tree blow-up and comparison conjecture for KSBA moduli

Let $M_N$ be the KSBA moduli stack of index one covers, let $M_N^{\operatorname{Bub}}$ be the moduli stack of bubble-tree index one cover DM stacks, and let $M_N^{\operatorname{lci}}$ be the moduli stack of lci covers. The relevant locus in $M_N$ is the locus whose corresponding surfaces contain bad simple elliptic or cusp singularities with embedded dimension at least $6$. Bubble-tree blow-up and comparison conjecture. The moduli stack $M_N^{\operatorname{Bub}}$ can be obtained from $M_N$ by finitely many blow-ups along this locus. Moreover, there is a birational morphism

$$M_N^{\operatorname{Bub}}\dashrightarrow M_N^{\operatorname{lci}}$$

for which both stacks admit contraction morphisms to $M_N$.

This conjecture predicts that the bubble-tree compactification is birationally related to the lci-cover compactification and is obtained from the KSBA space by resolving the specified singularity locus. Its status is not established in the supplied text.