The linear cycle-and-edge cover conjecture for 2-regular graphs

Let $G$ be an $n$-vertex graph. A cycle-and-edge cover is a cover of the edge set of $G$ by subgraphs that are $2$-regular graphs or single edges. The linear cycle-and-edge cover conjecture. For all $n$-vertex graphs $G$, the minimum number of such subgraphs satisfies

$$\operatorname{f_{re}}(G)=O(n).$$

This is presented as a weakened version of the paper's main conjecture, replacing decompositions into cycles and edges by decompositions into $2$-regular graphs and edges. The supplied text does not state a resolution.