The crown graph extremal representation-number conjecture

Let $H_{n,n}$ be the crown graph obtained from the complete bipartite graph on two parts of size $n$ by deleting a perfect matching. The crown graph extremal conjecture. $H_{n,n}$ has the highest representation number among all bipartite graphs on $2n$ vertices. The representation number of $H_{n,n}$ is known to be $4\lceil n/2\rceil$ for $n\geq 5$, but it remains unknown whether it is maximal among all bipartite graphs on $2n$ vertices.