The first threshold conjecture for unbounded indecomposable permutations

Let $\mathcal{C}$ be a sum-closed permutation class, and let $\mathcal{C}^{\not\oplus}$ denote its indecomposable permutations. Define $\tau$ to be the unique positive root of

$$x^3-3x^2+2x-1=0,$$

so that $\tau\approx 2.32472$. First threshold conjecture. If $|\mathcal{C}^{\not\oplus}_n|$ is unbounded, then

$$\operatorname{gr}(\mathcal{C})\geq\tau;$$

in other words, $\tau$ is the smallest possible growth rate of a sum-closed class with unbounded numbers of indecomposable permutations. The class $\operatorname{Av}(321,3142,2413)$ realizes growth rate $\tau$, while the paper establishes that the threshold is at least $\xi\approx2.30522$; determining whether no smaller value than $\tau$ is possible remains open.