**Conjecture**Let be a -connected cubic graph and let be a -regular subgraph such that is connected. Then has a cycle double cover which contains (i.e all cycles of ).

Used definitions in the above conjecture: a "cycle" is a connected 2-regular subgraph, a "cycle double cover" of a graph is a set of cycles of such that every edge of 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).