Fuller–Gould conjecture on dense $K_4^-$-saturated graphs

Let $K_4^-$ be the graph obtained from the complete graph $K_4$ by deleting one edge. A graph is $K_4^-$-saturated if it contains no copy of $K_4^-$, but adding any missing edge creates at least one copy of $K_4^-$. Write $[a,b]=\{a,a+1,\ldots,b\}$.

Fuller–Gould conjecture. The $K_4^-$-saturated graphs with sizes in the interval

$$\left[\left\lfloor\frac{n}{2}\right\rfloor\left\lceil\frac{n}{2}\right\rceil-n+7,\left\lfloor\frac{n}{2}\right\rfloor\left\lceil\frac{n}{2}\right\rceil\right]$$

are of two types: complete bipartite graphs with partite sets of nearly equal size, and 3-partite graphs with two partite sets of nearly equal size and one partite set of order one.

This conjecture describes the structure of $K_4^-$-saturated graphs whose numbers of edges are close to the extremal value $\left\lfloor n/2\right\rfloor\left\lceil n/2\right\rceil$. The cited work establishes the preceding range of edge sizes up to one below the interval's lower endpoint, while the proposed structural classification for this dense range is not resolved in the supplied source.