Hamilton-cycle reconstruction conjecture from induced subgraphs

For a graph $G$, let $[?]$ denote the multiset of its induced subgraphs on $[?]$ vertices. There are constants $0\leq c<1$ and $n_0\geq 0$ such that, for every integer $n\geq n_0$, the number of Hamilton cycles of every $n$-vertex graph $G$ is uniquely determined by the multiset of induced subgraphs of $G$ on $\lfloor cn\rfloor$ vertices.

Hamilton-cycle reconstruction conjecture. There are constants $0\leq c<1$ and $n_0\geq 0$ such that, for all integers $n\geq n_0$, the number of Hamilton cycles of an $n$-vertex graph $G$ is uniquely determined from $\mathcal{C}(G,\lfloor cn\rfloor)$.

The conjecture proposes that Hamilton-cycle counts, despite involving connected spanning structures, can be reconstructed from induced subgraphs of a fixed linear-size fraction strictly smaller than the full graph. The supplied text gives no resolution, so the conjecture is treated as open.