Directed Borodin–Kostochka conjecture for maximum out-degree

Let $D$ be a digraph with maximum out-degree $\Delta^+(D)$, biclique number $\overset{\leftrightarrow}{\omega}(D)$, and dichromatic number $\vec{\chi}(D)$. Write $\vec{C_3}\boxplus \overleftrightarrow{K}_{\Delta-2}$ for the directed-cycle obstruction described above. Maximum-out-degree directed Borodin–Kostochka conjecture. If $\Delta^+(D)=\Delta\geq 9$ and $\overset{\leftrightarrow}{\omega}(D)\leq\Delta-1$, then

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

unless $D$ contains $\vec{C_3}\boxplus \overleftrightarrow{K}_{\Delta-2}$. This conjecture is presented as a stronger-parameter analogue of the preceding directed conjecture; the source proves the corresponding result for sufficiently large maximum out-degree but leaves the conjecture open.