Finitely many coloring graphs determine the graph

For a finite graph $G$, let $\mathcal C_k(G)$ be its coloring graph. Finite coloring-graph determination conjecture. There exists a function $f\colon \operatorname{Graphs}\to\mathbb N$ mapping finite graphs to natural numbers such that, for any graph $G$, the collection

$$\{\mathcal C_k(G)\}_{k=1}^{f(G)}$$

uniquely determines $G$.

This is presented as a seemingly stronger finite version of the coloring-graph completeness conjecture: because graphs are finite, the authors expect finitely many colors to suffice. The paper does not resolve whether such an $f$ exists.