Salia's cycle-cover conjecture for infinite bipartite graphs

Let $G$ be a bipartite graph with sides $A$ and $B$. For a subset $X\subseteq V(G)$ with $|X|\geq 2$, let $N^2_G(X)$ be the set of vertices having at least two neighbors in $X$. The graph $G$ has the double Hall property if $|N^2_G(X)|\geq |X|$ for every $X\subseteq A$ with $|X|\geq 2$.

Salia's conjecture. If $G$ has the double Hall property, then for every $X\subseteq A$ with $|X|\geq 2$ there is a cycle $C_X$ in $G$ such that

$$V(C_X)\cap A=X.$$

The conjecture proposes that the double Hall condition is sufficient to find a cycle covering any prescribed finite or infinite subset of $A$ of size at least two, with no other vertices of $A$ on the cycle. The supplied text gives no resolution, so the conjecture is recorded as open.