The 2-wqo one-point-extension conjecture for permutation classes

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{+1}$ denote its one-point extension. The 2-wqo one-point-extension conjecture. If $\mathcal{C}$ is 2-wqo, then $\mathcal{C}^{+1}$ is wqo. The assertion would follow from Pouzet's 2-wqo conjecture, since the proof in the source shows that every 4-wqo class has a wqo one-point extension; it also follows from the stronger conjecture that every 2-wqo class is lwqo.