Existence of exponential growth rates for hereditary ordered graph properties

Let $\mathcal{P}$ be a hereditary property of ordered graphs, and suppose there is a real constant $c$ such that $|\mathcal{P}_n|<c^n$ for every $n\in\mathbb{N}$.

Exponential growth-rate conjecture. The limit

$$\lim_{n\to\infty}(|\mathcal{P}_n|)^{1/n}$$

exists.

The claim asks whether every exponentially bounded hereditary property of ordered graphs has a well-defined exponential growth rate. The supplied text presents it as an open question and gives no resolution.