Goddyn's conjecture on cycle double covers

Let $G$ be a bridgeless graph, and let $C$ be a cycle in $G$. A cycle double cover is a list of cycles in $G$ in which each edge appears exactly twice.

Goddyn's conjecture. There exists a cycle double cover of $G$ containing $C$.

The paper says its proof resolves this stronger conjecture for 3-regular graphs, while the conjecture remains unresolved for graphs that are not 3-regular. Thus the general statement remains open.