De Simone–Körner conjecture on hereditary normality of graphs

Let $G$ be a graph. A graph is hereditarily normal if every induced subgraph of $G$ is normal, where a graph is normal when it has coverings by cliques and stable sets such that every clique in one covering intersects every stable set in the other. An induced subgraph is a 5-cycle or 7-cycle when it is isomorphic to the corresponding cycle graph.

De Simone–Körner conjecture. A graph $G$ is hereditarily normal if and only if neither $G$ nor its complement contains a 5-cycle or a 7-cycle as an induced subgraph.

The conjecture aims to characterize hereditary normality by the two known minimal obstructions, the 5-cycle, the 7-cycle, and the complement of the 7-cycle. The source gives no resolution status.