Hardness radius of Hamiltonian Cycle under edge-edit distance

Determine the hardness radius of Hamiltonian Cycle under the edge-edit distance, where the distance between two graphs on the same vertex set is the number of edges in their symmetric difference.

Background

The paper studies Sidestep(Hamiltonian Cycle, dist, d), which permits replacing an input graph by a graph differing in at most d(n) edge edits and then returning a correct Hamiltonian-cycle answer for the modified graph. The authors establish that the problem is NP-hard for radius n^{1/2-o(1)} and is polynomial-time solvable for radius n/3. Consequently, the exact hardness radius remains unresolved between these bounds.