The degree-four planar line-graph conjecture

Let $G$ be a graph, let $L(G)$ denote its line graph, and let $\Delta(G)$ denote its maximum degree. A graph is word-representable if it admits a word representation, and an orientation is $k$-semi-transitive when it satisfies the corresponding $k$-semi-transitivity condition. Degree-four line-graph conjecture. The line graph $L(G)$ of a graph $G$ with $\Delta(G)\leq 4$ is at least 3-semi-transitively orientable. Moreover, if $G$ is non-word-representable, $\Delta(G)\leq 4$, and $G$ is planar, then $L(G)$ is word-representable.

The claim proposes relationships between the semi-transitivity and word-representability of a graph and those of its line graph, in the degree-four planar setting. The source presents these as possible relationships rather than established results; their general validity is not resolved there.