The smallest speed above 2 conjecture

Let $\mathcal{S}$ denote the set of exponential growth constants arising from hereditary properties of ordered graphs. The conjecture identifies the first member of this set strictly greater than $2$.

Smallest-speed-above-2 conjecture. The smallest $c\in\mathcal{S}$ with $c>2$ is the largest real root of

$$x^5=x^4+x^3+x^2+2x+1,$$

and is approximately $2.03$.

The paper gives an example of a hereditary property whose speed realizes this candidate value. The supplied text gives no proof or later resolution of the identification.