Acyclic system conjecture on extendability of boundary subdivisions of nΔ₃

Let $n\Delta_3$ be the dilation by $n$ of the standard $3$-simplex. A fine mixed subdivision structure on the boundary facets is boundary data arising from a fine mixed subdivision, and an acyclic system of permutations on the edges is one with no cyclic pattern on the edges. A 33-cyclic pattern is the forbidden pattern defined in the paper, and linkage is the matching-ensemble compatibility condition. Acyclic system extendability conjecture. Given $n\Delta_3$, suppose that its four boundary facets carry a fine mixed subdivision structure arising from an acyclic system of permutations on the edges. Then this boundary information can be completed to a fine mixed subdivision of $n\Delta_3$ if and only if linkage holds and there are no 33-cyclic patterns. The preceding corollary establishes the analogous assertion for $4\Delta_3$; the all-$n$ statement is proposed as a further direction and is unresolved in the supplied text.