The polyhedral semistable reduction conjecture

Let $f:\Delta_X\rightarrow\Delta_B$ be a surjective morphism of rational conical polyhedral complexes such that $f^{-1}(0)=\{0\}$. A projective subdivision replaces the complexes by projective subdivisions inducing a morphism, and a lattice alteration replaces the lattices by compatible sublattices and induces the corresponding morphism. A morphism is semistable when both complexes are nonsingular and, for every cone $\sigma\in\Delta_X$, one has $f(\sigma)\in\Delta_B$ and $f(N_\sigma)=N_{f(\sigma)}$. The polyhedral semistable reduction conjecture. There exists a projective subdivision $f':\Delta_X'\rightarrow\Delta_B'$ followed by a lattice alteration $f”:\Delta_X”\rightarrow\Delta_B”$ such that $f”$ is semistable. This is the combinatorial formulation underlying semistable reduction. The supplied text does not state a resolution status for this formulation; the paper’s introduction says that the corresponding geometric problem is solved for relative dimension at most three.