Lovász–Woodall conjecture on cycles through prescribed independent edges

Let $G$ be a $k$-connected graph, and let $S$ be a set of $k$ independent edges of $G$. Here, independent edges have no common endpoint.

Lovász–Woodall conjecture. If $k$ is even or $G-S$ is connected, then $G$ contains a cycle containing every edge in $S$.

This conjecture is part of the classical problem of finding cycles through prescribed edges. The supplied source does not state whether it has been resolved.