Borodin's cyclic coloring conjecture

Let $G$ be a connected plane graph with vertex set $V(G)$, edge set $E(G)$, and face set $F(G)$. A cyclic coloring assigns distinct colors to any two different vertices incident with the same face. Write $\chi_c(G)$ for the minimum number of colors in such a coloring, and let $\Delta^*(G)$ denote the maximum face degree of $G$.

Borodin's cyclic coloring conjecture. Every connected plane graph $G$ satisfies

$$\chi_c(G) \leq \left\lfloor \frac{3}{2}\Delta^*(G) \right\rfloor.$$

The lower bound $\chi_c(G) \geq \Delta^*(G)$ is immediate, and Borodin showed that the conjectured upper bound, when true, is best possible. It is known for maximum face degree at most $4$ and for maximum face degree $6$, and holds asymptotically up to an arbitrarily small additive factor, but remains open in general.