Ryjáček et al.'s line-graph minimum-degree conjecture

A line graph is a graph whose vertices represent the edges of another graph, with two vertices adjacent exactly when the corresponding edges share an endpoint. A graph is Hamiltonian if it contains a cycle through all its vertices, and $\delta(G)$ denotes its minimum degree.

Ryjáček et al.'s conjecture. Every $4$-connected line graph with minimum degree at least $5$ is Hamiltonian.

The source states that this conjecture is weaker than Thomassen's conjecture, which omits the minimum-degree condition. Its status is not resolved in the supplied text.