Martin's finite CRG conjecture for edit distance functions

Let $d4d8$ be a non-trivial hereditary property, and let $d4d2(d4d8)$ be the set of CRGs avoiding its forbidden graphs. For each $p a[0,1]$, let $g_K(p)$ denote the quadratic minimization function associated with a CRG $K$. Martin's finite CRG conjecture. For every $d700>0$, there is a finite \subset $d4d2'\subseteqd4d2(d4d8)$ such that

$$\operatorname{ed}_{\mathcal{H}}(p)=\min_{K\in\mathcal{K}'}g_K(p),\qquad\text{for all }p\in(\varepsilon,1-\varepsilon).$$

This conjecture asks whether the edit distance function is determined by finitely many CRGs uniformly on every compact subinterval of $(0,1)$.