Bensmail–Renault conjecture on locally irregular colorings of oriented graphs

For $a,b\in\{+,-\}$, a digraph is $(a,b)$-locally irregular if, for every arc $uv$, $d^a(u)\neq d^b(v)$; let $\mathrm{lir}^{(+,+)}(D)$ be the minimum number of colors in an arc coloring whose color classes are $(+,+)$-locally irregular. Bensmail–Renault conjecture. Every oriented graph $D$ satisfies

$$\mathrm{lir}^{(+,+)}(D)\leq 3.$$

The source reports that the conjecture is open, while a general upper bound of five is known for arbitrary digraphs and deciding whether the index is at most two is NP-complete.