Edge-count threshold conjecture for rigidity in binomial random graphs

Let $G\sim G(n,p)$ be a binomial random graph, where $G(n,p)$ contains each edge independently with probability $p$. Write $\textbf{whp}$ for “with high probability,” meaning with probability tending to $1$ as $n\to\infty$, and let $d$ be the rigidity dimension.

Random-graph rigidity conjecture. For every $p=\omega(\log n/n)$, $G\sim G(n,p)$ is $\textbf{whp}$ $d$-rigid for

$$d=(1-o(1))(1-\sqrt{1-p})n.$$

This conjecture proposes that, once the logarithmic connectivity threshold is passed, the edge-count obstruction is asymptotically the bottleneck for rigidity. The paper establishes results only up to logarithmic factors, leaving the asserted sharp threshold open.