Majority distinguishing edge coloring

We consider edge colorings of graphs. An edge coloring is a majority coloring if for every vertex at most half of the edges incident with it are in one color. And edge coloring is a distinguishing coloring if for every non-trivial automorphism at least one edge changes its color. We consider these two notions together. We show that every graph without pendant edges has a majority distinguishing edge coloring with at most $\lceil\sqrt{\Delta}\rceil+5$ colors. Moreover, we show results for some classes of graphs and a~general result for symmetric digraphs.