The nowhere-dense conjecture for subthreshold permutation-class growth rates

Let $\lambda$ be the unique real root of $x^5-2x^4-2x^2-2x-1$, and let $S$ be the set of growth rates of permutation classes that are below $\lambda$. Nowhere-dense growth-rate conjecture. The set $S$ is nowhere dense. The source gives no resolution of this conjecture; it is intended to assert that, despite all real numbers above the threshold being growth rates, the subthreshold growth rates have no interval in their closure.