On the Turánability and tileability of oriented graphs

An oriented graph $H$ is Turánable (resp. tileable) if there exist $n_0\in \mathbb{N}$ such that every semi-regular near-tournament on $n\ge n_0$ vertices contains a copy of $H$ (resp. a perfect $H$-tiling). We disprove a conjectured characterization of Turánable oriented graphs by DeBiasio, Han, Lo, Molla, Piga, and Treglown, show that there are Turánable oriented graphs which are not tileable, and provide a new example of tileable oriented graph.