The acyclic high-degree induced-subgraph conjecture for facial unique-maximum colorings

Let $G$ be a plane graph, let $d(v)$ denote the degree of a vertex $v$, and define

$$X=\{v\in V(G):d(v)\geq 4\}.$$

Let $\chi_{\mathrm{fum}}(G)$ be the minimum number of colors in a proper facial unique-maximum coloring of $G$.

Acyclic high-degree conjecture. If the induced subgraph $G[X]$ is acyclic, then

$$\chi_{\mathrm{fum}}(G)\leq 4.$$

The paper proves the result when $G[X]$ is a star forest and proposes extending it to all acyclic induced subgraphs; the conjecture remains open in the supplied text.