The isolated-vertices conjecture for logconcave random graphs

Let $F$ be a distribution in the positive orthant with a down-monotone logconcave density. For each coordinate $e$, write ${\sf E}(X_e^2)=\sigma_e^2$, and let $\sigma_{\min}=\min_e\sigma_e$ and $\sigma_{\max}=\max_e\sigma_e$. Let $G_{F,p}$ denote the random graph generated from $F$ with parameter $p$. The isolated-vertices conjecture. There exists a constant $c_0$ such that, if

$$p < \frac{c_1\sigma_{\min}\ln n}{n},$$

then $G_{F,p}$ has isolated vertices with high probability. The conjecture concerns the gap between the theorem's lower connectivity threshold of order $\sigma_{\min}/n$ and its upper threshold of order $\sigma_{\max}\ln n/n$; the source notes that it was incorrectly claimed as a theorem in an early version, while its resolution is not established in the supplied text.