Clique–Stable Set separation conjecture for graphs

Let $G$ be a graph on $n$ vertices. A clique–stable set separator is a family of cuts such that every disjoint clique and stable set of $G$ are separated by one of the cuts. Clique–Stable Set separation conjecture. There is a polynomial $Q$ such that every graph $G$ on $n$ vertices has a CS-separator of size at most

$$Q(n).$$

The source says that this positive formulation is generally believed to be false, while establishing such separators for several graph classes. Its general status is not resolved in the supplied text.