The coloring-graph family is a complete graph invariant

For a finite graph $G$, let $\mathcal C_k(G)$ be its coloring graph for $k\geq 1$, and let $\pi_G^{(H)}$ count induced copies of a graph $H$ in these coloring graphs. Coloring-graph completeness conjecture. For any graph $G$, the collection

$$\{\mathcal C_k(G)\}_{k\geq 1}$$

uniquely determines $G$. Equivalently, graphs $G_1$ and $G_2$ are isomorphic if and only if $\pi_{G_1}^{(H)}=\pi_{G_2}^{(H)}$ for every graph $H$.

The conjecture asserts that the full family of coloring graphs, or equivalently all $H$-polynomials, is a complete invariant, despite discarding the individual coloring data. The paper presents it as open.