Super-neighborhood property implies a cycle covering the distinguished bipartition

Prove that every bigraph G=(X,Y) satisfying the super-neighborhood property contains a cycle containing all vertices of X, for arbitrary |X|≥3.

Background

A bigraph G=(X,Y) is said to satisfy the super-neighborhood property if every subset S⊆X with |S|≥3 has at least |S| vertices in its super-neighborhood N²(S), and the induced subgraph G[S∪N²(S)] is 2-connected. Every supercyclic bigraph satisfies this property, and the conjecture asks whether the converse holds.

The paper proves the conjecture for 3≤|X|≤7, but does not resolve the general case. A counterexample would be a critical bigraph, so the conjecture is also described in the paper as asserting that no critical bigraph exists.