\begin{conjecture} Let $G$ be a $2$-connected cubic graph and let $S$ be a $2$-regular subgraph such that $G-E(S)$ is connected. Then $G$ has a cycle double cover which contains $S$ (i.e all cycles of $S$). \end{conjecture}

Used definitions in the above conjecture: a "cycle" is a connected 2-regular subgraph, a "cycle double cover" of a graph $G$ is a set of cycles of $G$ such that every edge of $G$ is contained in precisely two cycles of the set. This conjecture has been motivated by Theorem 3, respectively, Theorem 4 in www.arxiv.org/abs/1711.10614. A weaker conjecture (Conjecture 14) has been stated in "Snarks with special spanning trees" (see www.arxiv.org/abs/1706.05595).