Amiable colorings of 3-row graphs

A 3-row graph $R$ has vertices $v_{ij}$, where $1 \leq i \leq 3$ and $1 \leq j \leq s$, with each column $C_j:=\{v_{1j},v_{2j},v_{3j}\}$ independent. Let $R_C$ be the graph obtained by identifying all vertices in each column $C_j$ to a single vertex $c_j$. An amiable coloring of $R$ is a pair consisting of a $3$-vertex-coloring and a $3$-edge-coloring satisfying the three conditions defined in the paper: distinct vertices in each column receive distinct colors; no vertex color equals the color of any incident edge; and, for every $i \in \{1,2,3\}$ and $j \in \{1,2,\ldots,s\}$, the sum of the numbers of color-$i$ edges incident with the three vertices in column $j$ is even.

Amiable-coloring conjecture. If $R$ is a $3$-row graph such that $R_C$ is eulerian, then $R$ admits an amiable coloring.

This conjecture is introduced as a coloring problem intended to yield a $6$-cycle double cover for the original cubic graph. The source presents it as a route toward proving the cycle double cover conjecture for cubic graphs with Kotzig-frames; its resolution is not supplied here.