The characterization conjecture for 2-extremal digraphs

Let $\mathcal H_2$ be the smallest class of digraphs containing symmetric odd cycles and closed under taking directed Hajós joins and $2$-Hajós tree joins. A digraph is $2$-extremal when it has the relevant extremal colouring property defined in the paper. Characterization conjecture. A digraph is $2$-extremal if and only if it belongs to $\mathcal H_2$. The paper observes that digraphs in $\mathcal H_2$ are $2$-extremal; the converse, which would characterize all $2$-extremal digraphs, remains conjectural.