Pouzet's 2-wqo conjecture for permutation classes

For a permutation class $\mathcal{C}$ and an integer $n\geq 1$, say that $\mathcal{C}$ is $n$-well-quasi-ordered (or $n$-wqo) if the set of permutations in $\mathcal{C}$ labeled by an $n$-element antichain is well-quasi-ordered. Pouzet's conjecture. A permutation class $\mathcal{C}$ is 2-wqo if and only if it is $n$-wqo for every $n\geq 1$. The conjecture is open for permutation and graph classes, and more generally in the relational-structure setting in which Pouzet posed it.