Component-size conjecture for random graphs locally modelled on lattices

Let $\mathbb{L}^d$ denote the $d$-dimensional lattice. Define

$$r_0(2)=2,\qquad r_0(d)=3\quad\text{for all }d\geq 3.$$

Let $d\in\mathbb{N}$ with $d\geq 2$, and let $r\in\mathbb{N}$ with $r\geq r_0(d)$. Component-size conjecture. With high probability, the largest component of $G_n(\mathbb{L}^d,r)$ has order

$$\Theta_{d,r}\!\left(n^{1/(d+1)}\log n\right).$$

This is proposed as a refinement of the preceding largest-component results, which give the corresponding phenomenon in a less precise or more general setting; the asserted asymptotic order remains conjectural.