Morris's asymptotic conjecture for FC-family thresholds

For integers $k\geqslant 2$ and $n$, let $FC(k,n)$ denote the threshold quantity for $k$-uniform families on an $n$-element ground set as defined in the paper, and let $\Theta$ have its usual asymptotic meaning. Morris's asymptotic conjecture.

$$FC(k,n)=\Theta(n^{k-2})$$

for all $k\geqslant 2$. This is presented among the paper's open conjectures and predicts the asymptotic order of the threshold for every uniformity.