Plummer–Toft conjecture on cyclic coloring of 3-connected plane graphs

Let $G$ be a $3$-connected plane graph, and let $\Delta^\star$ be its maximum face size. A cyclic coloring of $G$ is a vertex coloring in which any two vertices incident with the same face receive distinct colors. Plummer–Toft's conjecture. Every $3$-connected plane graph with maximum face size $\Delta^\star$ has a cyclic coloring with at most $\Delta^\star+2$ colors. The conjecture is known for $\Delta^\star=3$, $\Delta^\star=4$, and $\Delta^\star\ge 18$, and this paper proves the cases $\Delta^\star=16$ and $\Delta^\star=17$; the remaining values are not resolved here.