Directed Borodin–Kostochka conjecture with a biclique–directed-cycle obstruction

Let $D$ be a digraph, let

$$\widetilde{\Delta}(D)=\max_{v\in V(D)}\sqrt{d^-(v)d^+(v)},$$

let $\overset{\leftrightarrow}{\omega}(D)$ be the biclique number, and let $\vec{\chi}(D)$ be the dichromatic number. Write $\vec{C_3}\boxplus \overleftrightarrow{K}_{\Delta-2}$ for the digraph formed from a directed $3$-cycle and a complete bidirected graph on $\Delta-2$ vertices, with all arcs in both directions between the two parts. Directed Borodin–Kostochka conjecture. For every integer $\Delta\geq 9$ and every digraph $D$ satisfying $\widetilde{\Delta}(D)\leq\Delta$ and $\overset{\leftrightarrow}{\omega}(D)\leq\Delta-1$,

$$\vec{\chi}(D)\leq\Delta-1$$

unless $D$ contains $\vec{C_3}\boxplus \overleftrightarrow{K}_{\Delta-2}$. The obstruction explains why the direct directed analogue of Borodin–Kostochka fails, while the source proves the conjectured bound for sufficiently large parameters.