The $G(\rho)<1+2\rho$ conjecture for addable minor-closed classes

Let $\mathcal{G}$ be an addable, minor-closed class of graphs, let $G(z)$ be its exponential generating function, and let $\rho$ be the radius of convergence of $G(z)$. $G(\rho)<1+2\rho$ conjecture. One has

$$G(\rho)<1+2\rho.$$

If true, this would imply that the closure of the set of limiting probabilities contains at least four intervals, equivalently at least three gaps. The paper does not establish the inequality; the difficulty is relating $\rho$ and $G(\rho)$.