Liang's matching and node-disjoint link conjecture for subcubic bipartite graphs

Let $G=(S,T;E)$ be a bipartite graph. An $S$-link is a path of length $2$ whose two endpoints lie in $S$, and a family of $S$-links is node-disjoint if no two links share a node. A matching is a set of pairwise nonincident edges.

Liang's conjecture. If every node in $S$ has degree at most $4$ and every node in $T$ has degree at most $3$, then $G$ has a matching $M$ and a family $\mathcal{F}$ of node-disjoint $S$-links such that every node $v\in T$ of degree $3$ is covered by an edge in

$$M\cup\left(\bigcup_{P\in\mathcal{F}}P\right).$$

The conjecture was proposed as a tool toward antimagic edge labelings: it was known to imply that every $4$-regular graph is antimagic. It is proved in the source, so the conjecture is solved.