The apex crown graph extremal representation-number conjecture

Let $G_n$ be the graph obtained from the crown graph $H_{n,n}$ by adding an apex, that is, a vertex adjacent to every vertex of $H_{n,n}$. The apex crown graph extremal conjecture. $G_n$ has the highest representation number among all graphs on $2n+1$ vertices. The graph $G_n$ is the worst known word-representable graph in terms of representation number; its representation number is known to be $\lfloor(2n+1)/2\rfloor$, while the conjectured extremality remains open.