Evans-type conjecture for cartesian products with $K_2$

Let $G$ be a graph, let $k$ be an integer, and write $G\square K_2$ for the cartesian product of $G$ and the complete graph on two vertices. Write $\chi'(G)$ for the edge-chromatic number of $G$. Evans-type conjecture for cartesian products with $K_2$. If every precoloring of at most $k$ edges of $G$ can be extended to a proper $\chi'(G)$-edge coloring, then every precoloring of at most $k+1$ edges of $G\square K_2$ is extendable to a proper $(\chi'(G)+1)$-edge coloring of $G\square K_2$. The paper verifies this conjecture for trees, complete and complete bipartite graphs, and graphs with small maximum degree, while giving partial results for general regular graphs; the general assertion remains open.