We give an explicit finite counterexample to the Beneš conjecture on products of stabilizers of uniform partitions. The example has a ground set of size |E| = 14, a uniform partition with seven blocks of size 2, and a permutation φ satisfying U ∧ φ(U) = 0. The set (φ S(U))6 is transitive, but (φ S(U))11 ≠ Sym(E). Equivalently, in the graph-theoretic formulation, the associated simple 2-regular ordered 2-stage graph L has L5 externally connected, while L10 is not rearrangeable. Non-rearrangeability is certified by a cut of capacity 5 in a 10-step time-expanded network, while the corresponding demand requires 6 edge-disjoint crossings.
A. A. Raoui (Fri,) studied this question.