Prove the Hamilton-cycle reconstruction conjecture

Prove that there exist constants 0 ≤ c < 1 and n_0 ≥ 0 such that, for every integer n ≥ n_0, the number of Hamilton cycles in every n-vertex graph G is uniquely determined by the multiset of induced subgraphs of G on ⌊cn⌋ vertices.

Background

The conjecture concerns reconstructing a connected spanning-structure count from substantially smaller induced subgraphs than the n−1-vertex deck used in the known result of Tutte.

It is distinct from the paper’s proved matching-polynomial reconstruction theorem because Hamilton cycles are connected spanning structures rather than collections of disjoint edges or copies of a fixed graph.