Count the ways to add a K4

Let \( G\) denote a graph on \( n\) vertices and \( \lfloor n^2/4 \rfloor +1\) edges containing no \( K_4\). Erdös and Tuza [1] consider the largest integer \( m\), denoted by \( f(n)\), for which there are \( m\) edges \( e\) in the complement of \( G\) so that \( G + e\) contains a \( K_4\).

Conjecture (proposed by Erdös and Tuza [1])

\[ f(n)=(1+o(1))\frac{n^2}{16}.\]

Bibliography
1 P. Erdös, Some of my old and new combinatorial problems, Paths, flows, and VLSI-layout (Bonn, 1988), Algorithms Combin., 9, 35-45, Springer, Berlin, 1990.