The polynomial-to-factorial speed jump for hereditary properties of oriented graphs

Let $cmathcal{P}$ be a hereditary property of oriented graphs, and let $cmathcal{P}_n$ denote its members on $n$ vertices. Then either

$$|\mathcal{P}_n|=\Theta(n^k)$$

for some $k\in\mathbb{N}$, or

$$|\mathcal{P}_n|\geq F_n^*$$

for every $n\in\mathbb{N}$ with $n\neq4$.

Conjecture on oriented-graph speeds. This predicts a jump from polynomial to at least factorial speed for unlabelled hereditary properties of oriented graphs. The paper poses it as a possible extension of its tournament results, and no resolution is supplied.