The hereditary speed conjecture for ordered hypergraphs

Let $\mathcal{P}$ be a hereditary property of ordered hypergraphs. For every constant $c>0$, suppose there exists an $N=N(c)\in\mathbb{N}$ such that $|\mathcal{P}_N|>c^N$. The ordered-hypergraph speed conjecture. Then, for every $n\in\mathbb{N}$,

$$|\mathcal{P}_n|\geqslant\sum_{k=0}^{\lfloor n/2\rfloor}{n\choose 2k}k!=n^{n/2+o(n)}.$$

This conjecture would provide a common generalization of the stated results for ordered hypergraphs, partitions, ordered graphs, and ordered graphs excluding fixed complete and complete bipartite ordered graphs. Its status is not resolved in the source.