Erdős–Gallai linear cycle-and-edge decomposition conjecture

A decomposition of a graph is a partition of its edge set into subgraphs of the indicated types. Erdős–Gallai conjecture. Every $n$-vertex graph has a decomposition into $O(n)$ cycles and edges. The source says this is equivalent to the corresponding question for decomposing Eulerian graphs into $O(n)$ cycles and records it as open.