Giro et al.'s conjecture on reconstructing a positive fraction of points

Let $f:[n]\rightarrow\mathbb{R}$ be an arbitrary injective function, and let $\varepsilon>0$. For a random graph $G\sim G(n,p)$ with $p\geq(1+\varepsilon)/n$, a subset $V'\subseteq[n]$ is sought such that the restricted framework $(G[V'],f|_{V'})$ is globally rigid in $\mathbb{R}$.

Giro et al.'s reconstruction conjecture. For every $\varepsilon>0$ and every $p\geq(1+\varepsilon)/n$, with high probability there exists a subset $V'\subseteq[n]$ of size $|V'|=\Omega_{\varepsilon}(n)$ such that $(G[V'],f|_{V'})$ is globally rigid in $\mathbb{R}$.

This conjecture asks whether the threshold $1/n$ for the emergence of the giant 2-connected component is also the threshold for reconstructing a positive fraction of the points of any injective embedding. The paper proves a stronger near-spanning result at a suitable constant multiple of $1/n$, but the stated conjecture remains open.