The in-degree bound for arc-coloured digraph constructions

Let $D=(N,A_0\cup A_1)$ be an arc-coloured digraph. Suppose every triangle has the form

$$\{u,v,w\}\text{ with }(u,v),(v,w)\in A_0\text{ and }(u,w)\in A_1$$

or

$$\{u,v,w\}\text{ with }(u,v),(v,w)\in A_1\text{ and }(u,w)\in A_0,$$

and every node has exactly $d$ incoming arcs in each of $A_0$ and $A_1$. The in-degree bound conjecture. Then

$$d\leqslant \lvert N\rvert/7.$$

This would establish that $13n/14$ is best possible for the class of constructions described in the paper. The conjecture is presented as a first step toward the asymptotic minimum-regularity conjecture for maximal $(2,3)$-antichains, and no resolution is supplied.