  We consider the problem of extending partial edge colorings of cartesian
products of graphs. More specifically, we suggest the following Evans-type
conjecture: If $G$ is a graph where every precoloring of at most $k$ precolored
edges 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 $G \square K_2$. In this paper we verify that
this conjecture holds for trees, complete and complete bipartite graphs, as
well as for graphs with small maximum degree. We also prove versions of the
conjecture for general regular graphs where the precolored edges are required
to be independent.
