Hamilton cycles in digraphs of unitary matrices

A set S ⊆ V is called a q⁺-set (q^--set, respectively) if S has at least two vertices and, for every u ∈ S, there exists v ∈ S, v ≠ u such that N⁺ (u) ∩ N⁺ (v) ≠ ∅ (N^- (u) ∩ N^- (v) ≠ ∅, respectively). A digraph D is called s-quadrangular if, for every q⁺-set S, we have | ∪ { N⁺ (u) ∩ N⁺ (v) : u ≠ 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.