Common-card bound for bipartite and non-bipartite graphs

Let $G$ be a bipartite graph and $H$ a non-bipartite graph, both on $n$ vertices. A common card is an unlabeled vertex-deleted subgraph occurring in the decks of both graphs. Bipartiteness conjecture. For $n$ large enough, the number of common cards between $G$ and $H$ is at most $\lfloor \frac{n}{2}\rfloor+1$. The conjecture proposes the expected sharp lower bound on the number of cards needed to distinguish bipartite from non-bipartite graphs; the source does not state whether it has been resolved.