The 2-DIOMEGA-ORDERING NP-completeness conjecture

Let $T$ be a tournament, and let $\operatorname{\overrightarrow{\omega}}(T)$ denote the minimum, over all orderings of $V(T)$, of the clique number of the corresponding backedge graph. The 2-DIOMEGA-ORDERING conjecture. The problem of deciding whether $\operatorname{\overrightarrow{\omega}}(T)\leq 2$ is NP-complete. The cases $k=1$ and $k\geq 3$ are respectively polynomial and NP-complete; the source presents the case $k=2$ as the remaining question, but does not provide a resolution.