Finite CRG representation conjecture for edit distance functions

Let $\mathcal{H}$ be a nontrivial hereditary property. For every $\varepsilon>0$, there exists a finite set of colored regular graphs (CRGs) $\mathcal{K}'=\mathcal{K}'(\varepsilon,\mathcal{H})$ such that

$$\operatorname{ed}_{\mathcal{H}}(p)=\min\left\{g_K(p):K\in\mathcal{K}'\right\}$$

for all $p\in(\varepsilon,1-\varepsilon)$.

Finite CRG representation conjecture. For every nontrivial hereditary property $\mathcal{H}$ and every $\varepsilon>0$, the edit distance function $\operatorname{ed}_{\mathcal{H}}(p)$ is the pointwise minimum of the finitely many functions $g_K(p)$ associated with the CRGs in some finite set $\mathcal{K}'(\varepsilon,\mathcal{H})$, throughout $p\in(\varepsilon,1-\varepsilon)$.

This conjecture would show that, away from the endpoints $0$ and $1$, edit distance functions of hereditary properties can always be determined by finitely many colored regular graphs. The supplied text does not state whether the conjecture is known or open.