The lwqo consequence of wqo for one-point extensions of permutation classes

Let $\mathcal{C}$ be a permutation class, and let $\mathcal{C}^{+1}$ denote its one-point extension. A class is lwqo when it is well-quasi-ordered under labeling by arbitrary wqo label posets. The one-point-extension conjecture. If $\mathcal{C}^{+1}$ is wqo, then $\mathcal{C}$, and hence also $\mathcal{C}^{+1}$, is lwqo. This is motivated by the absence of known counterexamples and by the fact that lwqo is preserved between $\mathcal{C}$ and its one-point extension in the forward direction, while the converse for wqo is not known.