Daligault–Rao–Thomassé conjecture on well-quasi-ordering and clique-width

A graph class is hereditary if it is closed under taking induced subgraphs. Such a class is finitely defined if it can be characterised by a finite set of minimal forbidden induced subgraphs. A class is well-quasi-ordered by the induced subgraph relation if it contains no infinite antichain under that relation. The clique-width of a graph is the minimum number of labels needed to construct it using the standard clique-width operations.

Daligault–Rao–Thomassé conjecture. If a finitely defined hereditary class of graphs $\mathcal{G}$ is well-quasi-ordered by the induced subgraph relation, then $\mathcal{G}$ has bounded clique-width.

A negative answer is known for hereditary classes whose sets of minimal forbidden induced subgraphs are infinite, but the finitely defined case remains open according to the source.