Conjecture on generalised cycles and H-graphs

A hereditary graph class with finite distinguishing number is one whose distinguishing number, as defined in the source, is finite. The induced subgraph relation is the ordering by induced-subgraph containment. The source defines antichains of generalised cycles and generalised H-graphs from the constructions associated with its preceding results. Generalised cycles and H-graphs conjecture. A hereditary class with finite distinguishing number is well-quasi-ordered by the induced subgraph relation if and only if it contains only finitely many elements from every antichain of generalised cycles and generalised H-graphs. This is proposed as a characterization of well-quasi-ordering for hereditary classes with finite distinguishing number; the source does not state that either direction has been proved.