Circuit characterization of distance reduction for toric ideals of graphs

Let $G$ be a simple graph, let $A_G$ be its incidence matrix, let $I_G$ be the corresponding toric ideal, and let $M$ be a minimal Markov basis for $I_G$. A Markov basis is distance-reducing when it has the distance-reduction property for the fibers of $I_G$. Circuit characterization conjecture. $M$ is distance-reducing if and only if $M$ reduces the distance of the circuits of $A_G$. This conjecture asserts that, for toric ideals of graphs, it is enough to check distance reduction on circuits. It is proved in the paper when $I_G$ is a complete intersection toric ideal, while the general case remains open.