Generalized weak-subdivision conjecture for complete graphs

Let $G$ be a graph with $n$ vertices, and let $\overline{G}$ denote its complement. An induced weak-subdivision of $K_t$ is an induced subgraph of $G$ obtained by the weak-subdivision operation applied to the complete graph $K_t$; a bi-clique is a complete bipartite subgraph.

Generalized weak-subdivision conjecture. For every $\epsilon>0$ and integer $t\geq 3$, there exists $\delta>0$ such that, if $G$ has at most

$$\left(\frac{1}{t-1}-\epsilon\right)\frac{n^{2}}{2}$$

edges and does not contain an induced weak-subdivision of $K_t$, then $\overline{G}$ contains a bi-clique of size at least $\delta n$.

The conjecture generalizes the proved $K_5$ case discussed immediately beforehand and is stated as sharp in the source: examples formed from $t-1$ clique parts can have nearly the threshold number of edges while their complements have no linear-sized bi-clique. The supplied text does not indicate that the general case has been resolved.