Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains

It was conjectured in the paper “Stationary probability vectors of higher-order Markov chains” (Li and Zhang, 2015 [7]) that if the set of stationary vectors of the second-order Markov chain contains k  -interior points of the (k−1)(k−1)-dimensional face of the simplex ΩnΩn then every vector in the (k−1)(k−1)-dimensional face is a stationary vector. In this paper, we provide counterexamples to this conjecture.