Combinatorial semistable reduction for polyhedral maps

Let $f_\Delta:\Delta_X\to\Delta_B$ be a polyhedral map of rational conical polyhedral complexes, and assume

$$f_\Delta^{-1}(0)=\{0\}.$$

An alteration is a lattice alteration followed by a subdivision, and it is projective when the corresponding subdivision is projective. An induced alteration is the alteration obtained from the induced subdivision and lattice alteration described in the paper. A map of polyhedral complexes is semistable in the sense of the paper.

Polyhedral semistable reduction conjecture. There exists a projective alteration $\Delta_B^1\to\Delta_B$, with induced alteration $\Delta_X^1\to\Delta_X$, and a projective subdivision $\Delta_Y\to\Delta_X^1$, such that $\Delta_Y\to\Delta_B^1$ is semistable.

This is the combinatorial counterpart of semistable reduction and is intended to guide the corresponding geometric constructions. The supplied text states it as a conjecture and gives no resolution evidence.