The list chromatic index of simple graphs whose odd cycles intersect in at most one edge

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $\Delta(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(m\Delta(G):m)$-edge-choosable and $\Delta(G)$-edge-paintable for every $m \geq 1$.