Hamilton cycles in digraphs of unitary matrices

A set S ⊆ V is called an q-set (q−-set, respectively) if S has at least two vertices and, for every u ∈ S, there exists v ∈ S, v 6= u such that N(u) ∩ N(v) 6= ∅ (N−(u)∩N−(v) 6= ∅, respectively). A digraph D is called s-quadrangular if, for every q-set S, we have | ∪ {N+(u) ∩ N(v) : u 6= v, u, v ∈ S}| ≥ |S| and, for every q−set S, we have | ∪ {N−(u) ∩ N−(v) : u, v ∈ S)} ≥ |S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.