Common-neighborhood and common-antineighborhood conjecture for dense graphs

Let $G$ be a graph on $n$ vertices with edge density $\varepsilon$, meaning that $G$ has $\varepsilon\binom{n}{2}$ edges, up to the interpretation of the source's phrase “$\varepsilon$ fraction of the edges.” For a vertex $x$, let $N(x)$ be its neighborhood and let $\overline{N}(y)$ be the complement of the neighborhood of $y$. Common-neighborhood conjecture. There exist vertices $x,y\in V(G)$ and a subset $S\subseteq V(G)$ such that

$$|S|\geq \varepsilon(1-\varepsilon)n-O(1)$$

and

$$S\subseteq N(x)\cap\overline{N}(y).$$

The conjecture refines the preceding counting bound, which gives approximately $\sqrt{1-\varepsilon}-(1-\varepsilon)$ times $n$; for $\varepsilon=0.5$, the authors discuss improvements toward $0.25n$, with Paley graphs showing that $0.25+o(1)$ is best possible. The supplied text does not state that this conjecture has been resolved.