The regular linear cycle-and-edge cover conjecture

Let $G$ be an $n$-vertex $(2k+1)$-regular graph, where $k$ is a nonnegative integer. Let $\operatorname{f_{re}}(G)$ denote the minimum number of $2$-regular graphs and edges in a cover of the edges of $G$. The regular linear cycle-and-edge cover conjecture. For all $n$-vertex $(2k+1)$-regular graphs $G$,

$$\operatorname{f_{re}}(G)\leq n-1.$$

This is proposed as a special case of the preceding linear bound conjecture; Petersen's $2$-factor theorem is cited as reducing the regular case to odd-regular graphs. The supplied text gives no resolution, though it notes the claim is easy for odd-regular graphs with a perfect matching.