The three-state toric homogeneous Markov chain model has Markov degree two

We prove that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology, that a certain class of toric ideals are generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that the Markov degree was at most six.