Sharp-threshold conjecture for reconstructible subsets near the random-graph critical point

Let $V\subseteq\mathbb{R}$ with $|V|=n$, and let the known distances be distributed as in $G(V,\lambda/n)$ for a parameter $\lambda>0$. Call a subset reconstructible if every injection of $V$ into $\mathbb{R}$ preserving the distances on the known edges preserves all pairwise distances within that subset. Sharp-threshold conjecture. There exists $C>0$ such that, for every $\varepsilon>0$, if $\lambda=1+(C+\varepsilon)/\ln n$, then with high probability there exists a reconstructible subset of size $\Omega_{\varepsilon}(n/\ln^2 n)$, while if $\lambda=1+(C-\varepsilon)/\ln n$, then with high probability every reconstructible subset has size at most $o(n/\ln^2 n)$. This gives a quantitative formulation of the proposed phase transition in the critical window; neither side of the threshold is established in the source.