Polynomial-growth conjecture for permutations avoiding a fixed pattern

Fix a positive integer $t$. Let $\pi$ be a $k$-permutation that avoids at least one $t$-permutation, and let $L(\pi)$ denote its Stanley–Wilf limit.

Polynomial-growth conjecture. If $\pi$ avoids some $t$-permutation, then

$$L(\pi)=k^{O(1)}.$$

The paper presents this as a plausible conjecture. Its results show that containing all sufficiently large $t$-permutations forces super-polynomial growth in $k$, leaving the stated polynomial-growth regime as an open structural question.