Fabrici–Gö̈ring's unique maximal face-colour conjecture

Let $G$ be a plane graph. A proper colouring of $G$ assigns integers to its vertices so that adjacent vertices receive different integers. Fabrici–Gö̈ring's conjecture. Every plane graph has a proper colouring using the numbers $1$, $2$, $3$ and $4$ such that every face contains a unique vertex coloured with the maximal colour appearing on that face. This conjecture would strengthen the Four Colour Theorem; the cited source proposed it, while the paper proves the analogous statement with colours $1,\ldots,5$.