The no-accumulation-from-above conjecture for ordered graph speeds

Let $\mathcal{S}$ be the set of exponential growth constants arising from hereditary properties of ordered graphs. For each real number $c$, consider the interval immediately to its right.

No-accumulation-from-above conjecture. For every $c\in\mathbb{R}$, there exists $\varepsilon=\varepsilon(c)$ such that

$$\mathcal{S}\cap(c,c+\varepsilon)=\emptyset.$$

In particular, $\mathcal{S}$ has no accumulation points from above.

This is the proposed general “jump everywhere” principle for the possible exponential speeds. The supplied text gives no resolution.