Characterize graphs with uniquely quasirandom extremal constructions

Determine whether there exists an unweighted graph F whose extremal host graphs for the inducibility problem are all asymptotically quasirandom, in the sense that every extremal graph is quasirandom.

Background

The paper proves that a particular quantum graph has quasirandom extremal graphs, contrasting with the more usual complete multipartite or blow-up constructions in inducibility. It then asks whether the same phenomenon can occur for an ordinary, non-quantum graph. This is a concrete unresolved existence question about the possible structure of inducibility extremizers.