SOLVED

Let $G$ be a graph on $n$ vertices with diameter $2$, such that deleting any edge increases the diameter of $G$. Is it true that $G$ has at most $n^2/4$ edges?

A conjecture of Murty and Plesnik (see [CaHa79]) (although Füredi credits this conjecture to Murty and Simon, and further mentions that Erdős told him that the conjecture goes back to Ore in the 1960s). The complete bipartite graph shows that this would be best possible.

This is true (for sufficiently large $n$) and was proved by Füredi [Fu92].