Fabrici et al.'s four-color conjecture for facial unique-maximum edge-coloring

Let $G$ be a plane graph. Two distinct edges are facially adjacent if they are consecutive in some facial path, and a facial edge-coloring assigns distinct colors to facially adjacent edges. A FUM-edge-coloring is a facial edge-coloring in which each face is incident with a unique edge colored with the maximal color on that face. Write $\chi_{\mathrm{fum}}'(G)$ for the minimum number of colors in such a coloring.

Fabrici et al.'s conjecture. If $G$ is a $2$-edge-connected plane graph, then

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

The conjecture concerns the edge analogue of facial unique-maximum vertex-coloring. The paper proves the bound for $2$-connected plane graphs, but does not resolve the stated $2$-edge-connected version in full.