Erdős–Tuza conjecture on $K_4$-saturating edges

Let $G$ be an $n$-vertex $K_4$-free graph. An edge in $\overline{G}$ is $K_4$-saturating if adding it to $G$ creates a copy of $K_4$. Let $f(n,e)$ be the maximum integer $\ell$ such that every $n$-vertex $K_4$-free graph with $e$ edges has at least $\ell$ $K_4$-saturating edges.

Erdős–Tuza conjecture.

$$f(n,\lfloor n^2/4\rfloor+1)=(1+o(1))\frac{n^2}{16}.$$

Equivalently, every $n$-vertex $K_4$-free graph with $\lfloor n^2/4\rfloor+1$ edges has at least $(1+o(1))n^2/16$ $K_4$-saturating edges. This is an Erdős–Rademacher-type extremal problem: adding one edge beyond the Turán bound for $K_4$-free graphs should force quadratically many saturating non-edges. The supplied text gives no resolution, so the conjecture is recorded as open.