The Acyclic System Conjecture

Let $n\Delta_{d-1}$ be the $(d-1)$-simplex dilated by $n$. A system of permutations on its edges is a collection $\sigma=(\sigma_{ab})_{1\leq a\neq b\leq d}$, where each $\sigma_{ab}$ is a permutation of $[n]$ and $\sigma_{ab}$ is the reverse of $\sigma_{ba}$. For each $i\neq j$, direct the edge $ab$ from $a$ to $b$ when $i$ precedes $j$ in $\sigma_{ab}$. The system is acyclic if every resulting directed graph is acyclic.

Acyclic System Conjecture. Any acyclic system of permutations on the edges of the simplex $n\Delta_{d-1}$ is achievable as the system of permutations of a fine mixed subdivision.

The conjecture generalizes the characterization of lozenge tilings in dimension two. It was disproved by Francisco Santos, who constructed a counterexample for $n=5$ and $d=4$.