The sharp threshold conjecture for a linear-sized reconstructible set

Let $V$ be a set of $n$ points on the real line. A graph of known pairwise distances $(V,\mathcal{P})$ records which pairs of points have known distances; a set of vertices is reconstructible if its points can be determined from the known pairwise distances, up to the relevant ambiguities. For a random graph distributed as $\mathcal{G}(n,p)$, write whp for “with high probability.”

Sharp threshold conjecture. The threshold for the existence of a reconstructible set of size linear in $n$ is sharp and occurs at

$$p=\frac{1}{n}.$$

The preceding result establishes that $1/n$ is a weak threshold for a linear-sized reconstructible set, while the conjectured location agrees with the appearance of the giant component in $\mathcal{G}(n,p)$. The source presents this as a natural conjecture and gives no resolution.