Cartesian-product edge-precoloring conjecture for two graphs

Let $G$ and $H$ be graphs. Suppose every precoloring of at most $k < \chi'(G)$ edges of $G$ extends to a proper edge-coloring with \chi'(G) colors, and every precoloring of at most $l < \chi'(H)$ edges of $H$ extends to a proper edge-coloring with \chi'(H) colors. Let $G \square H$ denote their Cartesian product.

General Cartesian-product conjecture. Every precoloring of at most $k+l+1$ edges of $G \square H$ is extendable to a proper $(\chi'(G)+\chi'(H))$-edge-coloring of $G \square H$.

This is proposed as a far-reaching generalization of the preceding conjecture and would resolve numerous open problems about edge-precoloring extension in Cartesian products. The paper establishes initial cases, while the general statement remains open.