The double Hall path-cover conjecture

Let $G=(X,Y)$ be a dHp bigraph. Write $\mathsf{\Lambda}_G(X)$ for the neighborhood of $X$ in $Y$, and let a $Y$-$Y$ path be a path whose two endpoints lie in $Y$.

Double Hall path-cover conjecture. If

$$|\mathsf{\Lambda}_G(X)|\geq |X|+1,$$

then there exists a collection of disjoint $Y$-$Y$ paths whose union covers all of $X$.

The paper presents this as a conjecture weaker than Salia's cycle conjecture and proves that it is equivalent to another such weakening. Its general status is open.