The linear lower-bound conjecture for indecomposable permutations

Let $\mathcal{C}$ be a sum-closed permutation class, and let $\mathcal{C}^{\not\oplus}$ denote its indecomposable permutations. Linear indecomposable-count conjecture. If the sequence $|\mathcal{C}^{\not\oplus}_n|$ is unbounded, then

$$|\mathcal{C}^{\not\oplus}_n|\geq n-1$$

for every $n$. This stronger conjecture would imply the conjectured lower threshold $\tau$ for the growth rate of sum-closed classes with unbounded indecomposable counts. It remains open.