Lexicographic matchings cannot form Hamiltonian cycles

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is the union of two lexicographic matchings.