Exact-recovery phase-transition conjecture for logarithmic-degree Euclidean random graphs

For any $\lambda>0$, $d\in\mathbb{N}$, and $0\leq b<a\leq 1$, let $\nu_d$ denote the volume of the unit Euclidean ball in $d$ dimensions, and let $G_n\sim\mathcal{G}_n(\lambda n,a,b,d)$ be the graph model whose connection functions are $f_{in}^{(n)}(r)=a\mathbf{1}_{r\leq\log(n)^{1/d}}$ and $f_{out}^{(n)}(r)=b\mathbf{1}_{r\leq\log(n)^{1/d}}$. Exact-recovery phase-transition conjecture. If

$$\lambda\nu_d\left(1-\sqrt{ab}-\sqrt{(1-a)(1-b)}\right)>1,$$

then Exact-Recovery of $G_n$ is solvable. The preceding theorem establishes impossibility when the same expression is less than $1$, so this conjecture identifies the threshold for Exact-Recovery in this simplest logarithmic-degree instance of the model; the authors state that they do not pursue a proof.