Edge-precoloring conjecture for Cartesian products with balanced complete bipartite graphs

Let $G$ be an $r$-regular, triangle-free graph, and let $K_{n,n}$ be the complete bipartite graph with parts of size $n$. Let $G \square K_{n,n}$ denote their Cartesian product.

Balanced complete-bipartite Cartesian-product conjecture. If any precoloring of at most $k < r$ edges of $G$ can be extended to a proper $\chi'(G)$-edge-coloring of $G$, then any precoloring of at most $k+n$ edges of $G \square K_{n,n}$ can be extended to a proper $(\chi'(G)+n)$-edge-coloring of $G \square K_{n,n}$.

This conjecture is suggested as a generalization beyond the cases already known for stars, even cycles, and trees. Its resolution, or the analogous statement for $K_{n,m}$, would have implications for hypercube precoloring-extension problems; the conjecture remains open.