The marginal-edge probability conjecture for ERGMs

Let $X$ be sampled from the ERGM measure $\mu$ on graphs with vertex set $[n]$, let $e$ be any edge in $\binom{[n]}{2}$, and let $p^*$ denote the corresponding Erdős–Rényi edge probability. Marginal-edge probability conjecture. For any $e \in \binom{[n]}{2}$, the probability that $X \sim \mu$ contains $e$ is $p^* + O(n^{-1})$. The conjecture was resolved in the subcritical regime and is expected to be resolved in metastable wells in the supercritical regime by follow-up work; the general status is therefore not established here.