Growth-bound reconstruction conjecture for relational structures

Let $\mathcal{M}$ be a relational structure and let $G=\operatorname{Aut}(\mathcal{M})$. Let $F_n(G)$ denote the relevant orbit-counting sequence, and suppose that $F_n(G)$ is known together with a bound on its growth. Growth-bound reconstruction conjecture. Under these assumptions, $\mathcal{M}$ is reconstructible. The source presents this as a conjectural link between growth rates and reconstruction, without giving a resolution.