The algebraicity conjecture for hereditary ordered graph speeds

Let $\mathcal{S}$ be the set of exponential growth constants arising from hereditary properties of ordered graphs. Let $\mathbb{A}$ denote the set of algebraic numbers and $\mathbb{Q}$ the set of rational numbers.

Algebraicity conjecture. Every $c\in\mathcal{S}$ is either an integer or an irrational algebraic number; equivalently,

$$\mathcal{S}\subseteq\mathbb{Z}\cup(\mathbb{A}\setminus\mathbb{Q}).$$

The conjecture is motivated by the fact that all speeds found in the paper are integers or algebraic irrationals. The supplied text gives no resolution.