Tree–cycle decomposition cycle double cover conjecture

Let $G$ be a $2$-edge connected cubic graph with a decomposition into edge-disjoint subgraphs covering $E(G)$, consisting of a tree $T$ and a cycle $C$. Tree–cycle decomposition cycle double cover conjecture. The graph $G$ has a cycle double cover ${\cal S}$ with $C\in{\cal S}$. This is stated as a reformulation of the non-separating cycle double cover conjecture.