The zero-mutable log resolution conjecture

A zero-mutable log datum is a log datum on a rank-$2$ lattice that can be converted into a zero-mutable rank-one datum by a sequence of mutations; a zero-mutable log structure is a generic log structure subordinate to such a datum. Let $X=X_S$ and let $X^\dagger=(X,\mathfrak{M}_S)$ be the log scheme associated to a generic log structure subordinate to a zero-mutable log datum $S$.

Zero-mutable log resolution conjecture. The log scheme $X^\dagger$ admits a projective log crepant log resolution: there exist a log scheme $Y^\dagger$ and a projective morphism $\varepsilon^\dagger\colon Y^\dagger\to X^\dagger$ such that $Y=\bigcup_{i\in I}Y_i$, each $Y_i$ is irreducible and maps birationally to $X_i$; $Y^\dagger$ is log smooth over $\operatorname{Spec} k^\dagger$ outside finitely many points of the specified quotient-crossing type; the morphism is the identity away from the exceptional locus and preserves the log structures there; and $K_{Y^\dagger/k^\dagger}$ is $\varepsilon$-trivial.

This is the paper's main conjecture for the class of zero-mutable log structures. The introduction says that examples support it and relates it to work of Corti, Filip and Petracci and to the Gross--Siebert program; no resolution status is supplied in the source.