Vanishing proportion conjecture for 1-cross intersecting set pair systems

Let $m_n(*, *, 1)$ and $m_n(*, *, *)$ be the extremal quantities for the corresponding set pair systems, with notation as in the paper. Vanishing proportion conjecture.

$$\lim_{n\to \infty} \frac{m_n(*, *, 1)}{m_n(*, *, *)}=0.$$

The conjecture asserts that the extremal size in the 1-cross intersecting case is negligible compared with the unrestricted extremal quantity. The source compares this with related ratios tending to $1/2$ and presents the claim as an unresolved belief.