Conjectural abundance of pairs with large circle depth

Let $P$ be a set of $n$ points in the plane in general position, meaning that no three are collinear and no four are cocircular. A pair of points has the required circle depth when every circle through the pair has at least the specified number of points of $P$ both inside and outside. The implied abundance conjecture. Every such set $P$ contains $n+2$ pairs of points whose every circle has, both inside and outside, at least

$$\lfloor\tfrac{n}{4}\rfloor-1$$

points. This statement is presented as a direct implication of the preceding segment-depth conjecture, and is therefore open in the source.