Existence of a path of length n/2 in every bipartite graph with $d(A,B)=1/2$

Claim: Let $G=A\cup B$ be a balanced bipartite graph with $e(A,B)\ge n/2$ then G has a path of length n/2.

I know about the erdos-gallai theorem that would net a path of length n/4. By noting that $d(G)=2E(G)/V(G)\ge {n}^{2}/4n$

I suspect that the condition of being bipartite forces the ecistence of a longer path, and I am yet unaware of such a result or a counterexample.

Part of my intuition is from the fact that considering a disjoint union of 2 copies of ${K}_{n/4-1,n/4}$ which are edge-maximal biparite graphs not containing such a path, we then have these two subgraphs and two yet to be connected vertices on A, also:

$2e({K}_{n/4-1,n/4})=\frac{{n}^{2}}{8}-\frac{n}{2}<{n}^{2}/8$

And adding any edge would form a path of the desired length. Any help on how to go about proving this, or a reference for such a resukt would be greatly appreciated.