The upper-bound conjecture for strong majority edge-colorings of admissible graphs

Let $G$ be an admissible graph, and let ${\mathrm Maj'}(G)$ denote the minimum number of colors in a strong majority edge-coloring of $G$. Upper-bound conjecture.

If $G$ is an admissible graph, then

$${\mathrm Maj'}(G)\le 4.$$

The conjecture proposes a substantial improvement of the upper bound established in the preceding theorem for strong majority edge-colorings. The supplied text gives no resolution, so its status remains open.