Precoloring extension conjecture for Cartesian products with complete bipartite graphs

Let $G$ be a bipartite graph with maximum degree $\Delta(G)$, and let $K_{n,m}$ be a complete bipartite graph with $n\geq m$. Consider a precoloring of some edges of the Cartesian product $G\square K_{n,m}$ using at most $\Delta(G)+n$ colors, with the distance between every two precolored edges at least $2$.

Cartesian-product precoloring extension conjecture. This precoloring is extendable to a proper edge coloring.

This is proposed as a possible generalization of the paper's hypercube result and is presented as a question about whether the analogous theorem holds for Cartesian products with complete bipartite graphs. The source gives no resolution.