Asymptotic threshold conjecture for binary separating hash families

Let $N(w)$ denote the least $N$ such that there exists an $SHF(N;n,2,\{1,w\})$ with $n>N$.

Asymptotic threshold conjecture.

$$\lim_{w\rightarrow\infty}\frac{N(w)}{w^2}=1.$$

This conjecture predicts the quadratic asymptotic threshold for binary separating hash families with parameters $\{1,w\}$. The source provides no resolution.