The high-degree double Hall cycle conjecture

Let $k$ be a nonnegative integer, and let $G=(X,Y)$ be a dHp bigraph, where $|X|=n$. Here $\deg(y)$ denotes the degree of $y\in Y$.

High-degree double Hall cycle conjecture. If

$$n>\max\{2k+1,k(k+1)\}$$

and $\deg(y)\geq n-k$ for all $y\in Y$, then there is a cycle in $G$ covering all vertices of $X$.

This is a degree-restricted weakening of Salia's conjecture. The paper introduces it as one of two equivalent weaker conjectures, and the general assertion remains open.