Wong and Zhu's $(1,3)$-choosability conjecture for graphs without isolated edges

Let $G$ be a graph. A total-weight assignment is a proper assignment of real weights to the vertices and edges of $G$ such that the sums of incident edge weights and the vertex weight differ on every adjacent pair. The graph is $(k,k')$-choosable if every list assignment with lists of size $k$ on vertices and $k'$ on edges admits a proper total weighting from the lists.

Wong and Zhu's conjecture. Every graph with no isolated edges is $(1,3)$-choosable.

This is one of the list total-weighting conjectures proposed by Wong and Zhu; its resolution is not specified in the source.