The improved degree bound for spanning closed trails in edge-connected graphs

Let $G$ be a $k$-edge-connected graph, meaning every edge cut of $G$ has size at least $k$, with $k\geq 4$. A spanning closed trail is a closed trail containing every vertex of $G$. The improved spanning-trail conjecture. The graph $G$ admits a spanning closed trail meeting each vertex $v$ at most

$$\left\lceil\frac{d_G(v)-2}{k}\right\rceil+1$$

times. The source says that the preceding theorem can likely be improved to this simpler bound; no resolution is given.