The Galvin’s theorem is the generalized version of Dinitz conjecture that states that if the maximum degree of any bipartite graph is $Delta$, then its edges are colorable properly if each of its edges is given a list of $Delta$ colors to choose from.

The proof, as I have seen, is by using the fact that there exists a kernel perfect orientation of the line graph of bipartite graph. But, I have a method as follows: We know that even cycles are chromatic edge-choosable. The bipartite graphs consist of even cycles and possible chords (matchings). As each of the even cycle can be list edge colored with a list of exactly two colors, and the chords (matchings) can be list edge colored using a list having just one color, I think a suitable greedy list edge coloring ( by assigning a list of $Delta$ colors to each edge) would solve the problem.

What is the defect in the method above? Thanks beforehand.