Five-color conjecture for majority distinguishing edge colorings

Let $G$ be a graph of minimum degree at least $2$ that has a connected asymmetric spanning subgraph. A majority distinguishing edge coloring of $G$ assigns colors to the edges so that each color appears on at most half of the edges incident with every vertex, and only the identity automorphism of $G$ preserves the coloring.

Five-color conjecture. Every such graph $G$ has a majority distinguishing edge $5$-coloring.

The conjecture would improve the preceding general bound of seven colors for graphs with a connected asymmetric spanning subgraph. The source does not provide evidence that the conjecture has been resolved.