Expectation conjecture for mandatory and blocking edges in Erdős–Rényi graphs

Let $G_n$ be Erdős–Rényi graphs with parameters $(n,c/n)$, and let $\mathcal{M}_{\max}(G_n)$ be the set of maximum-size matchings of $G_n$. Let $\underline{\gamma}$ be the smallest solution of

$$x=e^{-ce^{-cx}},$$

and set $\overline{\gamma}=e^{-c\underline{\gamma}}$.

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

$$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{|E(G_n)|}\sum_{e\in E(G_n)}\mathbbm{1}_{\forall M\in\mathcal{M}_{\max}(G_n),\ e\in M}\right]=\underline{\gamma}^{2},$$ $$\lim_{n\to\infty}\mathbb{E}\left[\frac{1}{|E(G_n)|}\sum_{e\in E(G_n)}\mathbbm{1}_{\forall M\in\mathcal{M}_{\max}(G_n),\ e\notin M}\right]=(1-\overline{\gamma})^{2}.$$

This extends the corresponding law of large numbers from the subcritical regime to convergence in expectation, where concentration is not expected in general.