Matching-size conjecture for random geometric graphs

Let $X_1,\dots,X_n\stackrel{\mathrm{iid}}{\sim}\mathcal{P}_n$, where $r$ and $d$ may scale with $n$, and let $G(\{X_1,\dots,X_n\},r,d,\|\cdot\|)$ be the corresponding geometric graph. Let $M_r$ be its largest matching, and suppose

$$\mathbb{E}\left[|E(G(n,r,d,\|\cdot\|))|\right]=\binom{n}{2}\mathbb{P}\left(\|X_1-X_2\|_2^2<r^2\right)\underset{n\to\infty}{\longrightarrow}+\infty.$$

Matching-size conjecture. Under this assumption,

$$\mathbb{E}[|M_r|]=\Theta\left(\mathbb{E}[|E(G(n,r,d,\|\cdot\|))|]\wedge n\right),$$

and the same order holds with high probability for $|M_r|$. This would give the typical largest-matching size from the expected edge count, capped at the number of vertices, in a regime where the expected number of edges diverges.