Close the gap between upper and lower bounds for reconstructing arbitrary graphs from edge-deletion traces

Determine the tight sample complexity for reconstructing arbitrary n-vertex undirected graphs from unlabeled traces in the edge deletion model with p_v = 1 and p_e = 1/2 by closing the gap between the existing exp(Ω(n)) lower bound and known upper bounds, thereby establishing the precise exponential dependence on n.

Background

The authors prove an exp(Ω(n)) lower bound for distinguishing certain graphs under 1/2 edge-deletion noise, and they also provide a nontrivial distinguishing algorithm for a candidate hard pair using only exp(O(n^{1/3} log^{2/3} n)) traces, undermining a potential exp(Ω(n^2)) hardness construction.

They conjecture that exp(Ω(n^2)) traces are actually necessary in the worst case, but current techniques fall short; hence, identifying the exact complexity remains open.