Kostochka et al.'s super-neighborhood cycle conjecture for bigraphs

A bigraph is a bipartite graph $G=(X,Y)$ with an ordered vertex partition. For a set $S\subseteq X$, let $\mathsf{\Lambda}^2(S)$ be the vertices adjacent to at least two vertices of $S$. The graph is snp if, for every $S\subseteq X$ with $|S|\geq 3$, $|\mathsf{\Lambda}^2(S)|\geq |S|$ and $G[S\cup\mathsf{\Lambda}^2(S)]$ is $2$-connected.

Kostochka et al.'s conjecture. If a bigraph $G=(X,Y)$ is snp, then there is a cycle containing all vertices of $X$.

This is the graph-theoretic form of the conjectured equivalence between the super-neighborhood property and super-pancyclicity of hypergraphs. The conjecture is known for $|X|\in\{3,4,5,6,7\}$, while the general case remains open.