Supersaturation conjecture for ISP-systems

Let $N$ and $n$ be positive integers, and let $\mathcal{F}\subset \mathcal{P}(N)\times\mathcal{P}(N)$ have size

$$|\mathcal{F}|=\binom{n}{n/2}+x.$$

Assume that every $(A_j,B_j)\in\mathcal{F}$ satisfies $A_j\cap B_j=\emptyset$ and $|A_j|+|B_j|\leq n$. Supersaturation conjecture for ISP-systems. There are at least $\left(\left\lfloor\frac{n}{2}\right\rfloor+1\right)x$ pairs $(A_i,B_i),(A_j,B_j)\in\mathcal{F}$ for which $A_i\cap B_j=\emptyset$. This would provide the supersaturation needed to count intersecting set-pair systems, but the surrounding discussion explains that the proposed supersaturation result was not established and that the large-$N$ case presents an obstruction.