Distance-three matching conjecture for Cartesian powers of even cycles

Let $C_{2k}$ be an even cycle and let $C^d_{2k}$ denote the $d$-fold Cartesian product of $C_{2k}$ with itself. A distance-three matching is a matching whose distinct edges are at distance at least $3$, where edge distance is measured by the number of edges in a shortest path between endpoints. Distance-three matching conjecture. If $\varphi$ is an edge precoloring of a distance-three matching of $C^d_{2k}$, then $\varphi$ extends to a proper $4$-edge coloring of $C^d_{2k}$. The paper proves the analogous assertion for distance-four matchings using color interchanges, while the distance-three case remains open because the corresponding transformation cycles need not be edge-disjoint.