Two-color conjecture for strong local irregularity of digraphs

For a digraph $D$, define the balanced degree of a vertex $x$ by

$$\sigma(x)=d^+(x)-d^-(x).$$

A digraph is strongly locally irregular if $\sigma(x)\neq\sigma(y)$ for every arc $xy$, and let $\mathrm{slir}(D)$ be the minimum number of colors in an arc coloring whose color classes induce strongly locally irregular digraphs. Strong local irregularity conjecture. Every digraph $D$ satisfies

$$\mathrm{slir}(D)\leq 2.$$

The source notes that strong local irregularity is not directly controlled by the corresponding notion for orientations of simple graphs and presents the statement as an open conjecture supported by the paper’s investigation.