Compatible-extraction conjecture for zero mutable Laurent polynomials

Let $X_0$ be an affine Gorenstein toric $3$-fold defined by the cone over a polygon $\triangle$ at height $1$. For each edge $e$ of $\triangle$, let $\ell(e)$ be its lattice length. A zero mutable Laurent polynomial $f$ determines a decreasing partition $\mathbf{divstep}_e(f)$ of $\ell(e)$, and write $\mathbf{divstep}_e(f)^\vee$ for its dual partition. Let $\cup X_i$ be the toric degeneration associated with the central subdivision, and let $C_{ij}\subset X_i\cap X_j$ be the associated curves.

Compatible-extraction conjecture. If the partitions $(\mathbf p_e)_e$ are given by $\mathbf{divstep}_e(f)^\vee$, then there exists a compatible collection $Y_i\to X_i$ of divisorial extractions of $\cup C_{ij}\subset\cup X_i$.

Such a collection is the first step in constructing a log crepant log resolution and, ultimately, a smoothing of $X_0$. The paper proves this conjecture for the families called Tom, Jerry and Tyke, while the Spike cases are left as a conjectural part of the proposed method.