Ulam–Kelly reconstruction conjecture for graph homomorphism counts

Let $Hom(F,G)$ denote the number of graph homomorphisms from $F$ to $G$, and let $\mathcal{G}$ be the family of all graphs. For a family of graphs $\mathcal{F}$, say that $Hom(\mathcal{F},\cdot)$ is reconstructible if, for every graph $G$ with at least three vertices, the vector of homomorphism counts $Hom(\mathcal{F},G)$ is uniquely determined by the vectors $Hom(\mathcal{F},G\setminus i)$ for every vertex $i$ of $G$. Ulam–Kelly reconstruction conjecture. $Hom(\mathcal{G},\cdot)$ is reconstructible. This is the homomorphism-count formulation of the Ulam–Kelly graph reconstruction conjecture, asserting that a graph with at least three vertices is determined up to isomorphism by the collection of its vertex-deleted subgraphs. The source presents this formulation as a motivation and question; no resolution is given here.