Abramovich–Karu's polyhedral semistable reduction conjecture

Let $f\colon X \to B$ be a map of conical complexes. A map of conical complexes is semistable if $B$ is regular, every cone maps into a cone of $B$, the induced map on the relevant monoids is surjective, and $X$ is regular. Abramovich–Karu's polyhedral semistable reduction conjecture. There exists a projective alteration

$$b\colon B' \to B$$

and a projective subdivision

$$a\colon X'\to X\times_B B'$$

such that $f'\colon X' \to B'$ is semistable. This is the combinatorial counterpart of the geometric semistable reduction conjecture; its precise relation to the geometric problem is the subject of the paper.