General expectation conjecture for mandatory and blocking edges

Under the setting and notation of Theorem~, let $\mathcal{M}_{\max}(G_n)$ be the set of maximum-size matchings in $G_n$. Let $\zeta'$ be the probability law appearing in that setting, and let $k$ be the corresponding threshold parameter. For two independent samples $((i,Z),(i',Z'))$ from $\zeta'$, write $((i,Z),(i',Z'))\sim\zeta'\otimes\zeta'$.

General mandatory and blocking edge expectation conjecture. The expected proportions of edges belonging to every maximum-size matching and of edges belonging to no maximum-size matching satisfy

$$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{|E(G_n)|}\sum_{e\in E(G_n)}\mathbbm{1}_{\forall M\in\mathcal{M}_{\max}(G_n),\ e\in M}\right]=\mathbb{P}_{((i,Z),(i',Z'))\sim\zeta'\otimes\zeta'}(i+i'<k),$$ $$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{|E(G_n)|}\sum_{e\in E(G_n)}\mathbbm{1}_{\forall M\in\mathcal{M}_{\max}(G_n),\ e\notin M}\right]=\mathbb{P}_{((i,Z),(i',Z'))\sim\zeta'\otimes\zeta'}(i+i'>k).$$

This is the companion to the Erdős–Rényi conjecture and proposes expectation-level limits for generic offspring distributions beyond the regimes covered by the proved results. The definitions of $\zeta'$ and $k$ are delegated to the referenced theorem, so their precise hypotheses should be checked.