Distance-3 precoloring extension conjecture for graph products with an edge

Let $G$ be a graph with maximum degree $\Delta(G)$, and let $K_2$ be the complete graph on two vertices. Consider the Cartesian product $G\square K_2$. A precoloring is extendable if it extends to a proper edge coloring using the prescribed color set. Graph–edge product conjecture. If some edges of $G\square K_2$ are prescribed colors from a set of at most $\Delta(G)+1$ colors, and the distance between any two precolored edges is at least $3$, then the precoloring is extendable. The paper presents this as an open question concerning whether the preceding theorem holds for every graph $G$; the supplied text gives no evidence that it has been resolved.