Korpelainen et al.'s conjecture on finite definition and labelled well-quasi-ordering

Let $\mathcal{C}$ be a hereditary graph class, meaning that it is closed under taking induced subgraphs, and suppose that $\mathcal{C}$ is well-quasi-ordered under the induced subgraph relation. The class is labelled well-quasi-ordered if, for every well-quasi-order of labels, its labelled graphs contain no infinite antichain under the labelled induced subgraph relation. Korpelainen et al.'s conjecture. The class $\mathcal{C}$ is finitely defined if and only if it is labelled well-quasi-ordered. Every labelled well-quasi-ordered class is already known to be finitely defined, so the open part is the converse for well-quasi-ordered hereditary classes.