Non-separating cycle double cover conjecture

Let $G$ be a $2$-edge connected cubic graph. A cycle $C$ of $G$ is non-separating if $G-E(C)$ is connected. Non-separating cycle double cover conjecture. Every non-separating cycle $C$ of $G$ is contained in a cycle double cover ${\cal S}$ of $G$. This conjecture is presented as a motivation for the paper and remains open in the supplied text.