Georgakopoulos's Hamiltonian-circle conjecture for line graphs

Let $G$ be a locally finite graph, and let $L(G)$ denote its line graph. A Hamiltonian circle is a circle in the Freudenthal compactification of $L(G)$ that contains every vertex of $L(G)$. Georgakopoulos's conjecture. The line graph $L(G)$ of every locally finite $4$-edge connected graph $G$ has a Hamiltonian circle. This conjecture concerns the extension of Hamiltonian-cycle phenomena to infinite locally finite graphs through topological notions in their Freudenthal compactifications; the supplied passage gives no resolution, so its status remains open.