The extremal characterization of complete bidirected graphs without rainbow triangles

Let $D$ be an arc-colored digraph of order $n\geq 5$, meaning that the arcs of $D$ are colored. A rainbow triangle is a directed triangle whose three arcs have pairwise distinct colors; write $a(D)$ for the number of arcs and $c(D)$ for the number of colors used by $D$. Let $\overleftrightarrow{K}_{n}$ denote the complete bidirected graph on $n$ vertices.

Extremal characterization conjecture. If $D$ contains no rainbow triangles and

$$a(D)+c(D)=n(n-1)+\left\lfloor\frac{n^{2}}{4}\right\rfloor+1,$$

then

$$D\cong\overleftrightarrow{K}_{n}.$$

The paper proves the corresponding assertion for $n=3$ and $n=4$; the conjecture proposes that the same conclusion holds for every $n\geq 5$.