The maximum-degree-two high-degree 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 $G$.

Maximum-degree-two conjecture. If $G[X]$ has maximum degree at most $2$, then

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

The paper presents this as a stronger expected result following the matching case, but the supplied text gives no resolution; it remains open.