Threshold conjecture for the local pooling factor in random graphs

Let $G(n,r)$ be the random graph model under consideration, with edge-probability parameter $r^2$, and let $\operatorname{LoP}$ denote the graph property that the local pooling factor has the specified value. A function $f(n)$ is a threshold function for a graph property if the property changes from holding with probability tending to $0$ to holding with probability tending to $1$ as the parameter crosses the scale $f(n)$. Threshold conjecture for the local pooling factor. The function

$$r^2=1/n^{6/5}$$

is a threshold function for the graph property $\operatorname{LoP}$. The preceding corollary establishes a $0$-statement when $r^2\in o(1/n^{6/5})$; the conjecture asserts that the corresponding threshold scale also governs the $1$-statement.