DeBiasio–Han–Lo–Molla–Piga–Treglown conjecture on Turánable oriented graphs

Let $H$ be an oriented graph. For integers $a,b,c\geq 1$, let $D_{a,b,c}$ be the tournament obtained from the consistently oriented triangle $C_3$ by replacing its vertices with transitive tournaments on $a$, $b$, and $c$ vertices; write $D_s=D_{s,s,s}$.

DeBiasio–Han–Lo–Molla–Piga–Treglown conjecture. The oriented graph $H$ is Turánable if and only if there is an $s\in\mathbb{N}$ such that $H\subset D_s$.

The conjecture proposed a characterization of Turánable oriented graphs, motivated by the corresponding theorem for tournaments. It is refuted in the source paper: a Turánable oriented graph exists that is not a subgraph of $D_s$ for any $s\in\mathbb{N}$.