The proper abelian coloring conjecture for exceptional groups

Let $G$ be a bridgeless cubic graph. A proper $A$-coloring is a proper edge-coloring of $G$ by the non-zero elements of an abelian group $A$ such that the sum of the colors on the three edges incident with every vertex is zero. The exceptional abelian groups are $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$, $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$, $\mathbb{Z}_{10}$ and $\mathbb{Z}_{11}$. Proper abelian coloring conjecture. Every bridgeless cubic graph $G$ has a proper $A$-coloring for every exceptional abelian group $A$. This is the central open question left by the known classification of groups that do or do not properly color all bridgeless cubic graphs; a proper $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$-coloring is also known to imply proper colorings by the other exceptional groups.