Lozin–Razgon–Zamaraev's finite-definition conjecture for induced-subgraph well-quasi-ordering

Let $\mathcal{G}$ be a hereditary graph class defined by a finite set of forbidden induced subgraphs. A graph is labelled by assigning to each vertex an element of a quasi-order $(W,\leq)$. The class $\mathcal{G}$ is well-quasi-ordered by the labelled induced subgraph relation if, for every well-quasi-order $(W,\leq)$, it contains no infinite antichain of labelled graphs under that relation. Ordinary well-quasi-ordering by the induced subgraph relation is defined analogously without vertex labels.

Lozin–Razgon–Zamaraev's labelled induced-subgraph conjecture. If $\mathcal{G}$ is defined by a finite set of forbidden induced subgraphs, then $\mathcal{G}$ is well-quasi-ordered by the induced subgraph relation if and only if it is well-quasi-ordered by the labelled induced subgraph relation.

Labelled induced-subgraph well-quasi-ordering always implies ordinary induced-subgraph well-quasi-ordering, and Daligault, Rao and Thomassé proved that labelled well-quasi-ordering forces finite definability. The conjecture proposes that finite definability makes the converse implication hold; the supplied text does not establish its resolution.