Let $G$ be a subcubic graph.

Suppose that $G$ has an edge coloring $\varphi$ using colors from $\{1,2,3,4,5\}$ such that

- each edge is colored with a set of two elements from $\{1,2,3,4,5\}$ (e.g., $\varphi(e)=\{1,2\}$ for some edge $e$),
- if $e_1$ and $e_2$ are adjacent, then $\lvert\varphi(e_1)\cap \varphi(e_2)\rvert=1$,
- if the distance between two edges $e_1$ and $e_2$ is 2 (i.e, $e_1$ and $e_2$ are not adjacent, and there is an edge $e_3$ adjacent to both $e_1$ and $e_2$), then $\varphi(e_1)\neq \varphi(e_2)$.

Then, does $G$ have a 2-distance vertex 4-coloring (i.e., a proper vertex 4-coloring of $G$ such that every two vertices at distance 2 receive distinct colors)?

My feeling is that the answer is YES as no conterexample had been constructed yet.

Note that if $G$ admits a 2-distance vertex 4-coloring, then one can easily construct the above mentioned edge coloring.