Bubeck–Ding–Eldan–Rácz high-dimensional indistinguishability conjecture

Let $G(n,d,p)$ be the spherical random geometric graph and let $G(n,p)$ be the Erdős–Rényi random graph. For probability distributions on a finite set, define total variation distance by

$$\operatorname{TV}(\mu,\mu')=\frac{1}{2}\sum_{x\in\Omega}|\mu(x)-\mu'(x)|.$$

Write $a\gg b$ to mean that $a/b\to\infty$ as $n\to\infty$. Bubeck–Ding–Eldan–Rácz conjecture. If

$$d\gg n^3p^3(\log 1/p)^3,$$

then

$$\operatorname{TV}(G(n,p),G(n,d,p))\to 0$$

as $n\to\infty$. This is the high-dimensional, indistinguishable regime of the threshold problem for random geometric graphs. The source presents the statement as a conjecture and gives no resolution evidence for this exact formulation.