Consider the graph $G$ of order $n$ consisting of two disjoint cliques of even order $\frac{n}{2}=p+1$ (where $p$ is odd prime) joined by a bipartite graph (that is, deleting the edges of the two disjoint cliques from $G$ leaves a bipartite graph) of maximum degree $p$. Then, does the graph have list chromatic index $\le 2p+1$? The bipartite graph is also quite specific, in that it has one vertex in each partite set of degree exactly equal to $0,1,2,\dotsc,p$.

My view is that, by Schauz - Proof of the list edge coloring conjecture for complete graphs of prime degree paper, we have that the disjoint cliques are chromatic edge-choosable. In addition, the edges joining the two cliques is a bipartite graph, which is again chromatic edge-choosable by the Galvin's theorem. Thus, it makes me think the above question has a positive answer. By the way, the graph has chromatic index equal to $2p$, that is the graph is of class $1$. Any hints?