Hamiltonian-cycle conjecture for graphs of non-degenerate classical transportation polytopes

Let $P_c=\{x\mid Bx=c,\ x\geq 0\}$ be a non-degenerate $m\times n$ classical transportation polytope, and let its graph have the vertices and edges of $P_c$. Hamiltonian-cycle conjecture. The graph of every non-degenerate $m\times n$ transportation polytope has a Hamiltonian cycle when $mn>4$. This conjecture arises from enumeration data on transportation polytopes; the supplied source does not state a resolution, so its general status is open.