Malnegro–Ozeki's $\hat{A}$-coloring conjecture for line graphs of cubic graphs

Let $G$ be a 2-edge-connected simple cubic graph with an even number of edges. Its line graph $L(G)$ has one vertex for each edge of $G$, with two vertices adjacent when the corresponding edges of $G$ are incident. Let $\hat{A}$ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree-two vertex and the two adjacent degree-three vertices. An $H$-coloring of a graph is a pair of total functions $f:E(G)\to E(H)$ and $g:V(G)\to V(H)$ such that $f$ is a proper edge-coloring and, for every vertex $u$, $f(\partial_G(u))=\partial_H(g(u))$. Malnegro–Ozeki's conjecture. The line graph $L(G)$ admits an $\hat{A}$-coloring. The paper presents this as an analogue for future research after disproving the corresponding $\bar{X}$-coloring conjecture; no resolution is given here.