Minimal Markov basis conjecture for contingency-table margin equations

Let $C$ denote the set of small conditionals, let $J-1$ be the dimension of the underlying lattice, and consider the coefficient matrix of the Diophantine equation governing the margins. A minimal Markov basis conjecture asserts that, when $C \neq \emptyset$, this matrix has a Markov basis consisting of $J-1$ elements; equivalently, its corresponding toric ideal equals the lattice basis ideal. The assumption $C \neq \emptyset$ is necessary, since the paper gives an example where the claim fails for full conditionals. Supporting examples are included in the paper, but the conjecture is not resolved there.