Hajós' cycle decomposition conjecture for Eulerian graphs

Let $G$ be a simple Eulerian graph, meaning that every vertex of $G$ has even degree. A cycle decomposition of $G$ is a collection of cycles whose edge sets partition the edge set of $G$.

Hajós' conjecture. Every simple Eulerian graph $G$ has a cycle decomposition with at most

$$\left\lfloor \sfrac{(|V(G)| -1)}{2}\right\rfloor$$

many cycles.

The bound is motivated by Eulerian graphs containing a vertex of degree $|V(G)|-1$, which require at least $\left\lfloor \sfrac{(|V(G)|-1)}{2}\right\rfloor$ cycles in any decomposition. The source gives no resolution status; the conjecture is therefore recorded as open.