Polynomial Rödl property for graphs close to a proper hereditary family

Let $\mathcal{F}$ be a family of graphs. A graph $G$ on $n$ vertices is $\alpha$-close to $\mathcal{F}$ if one can add or delete at most $\alpha n^2/2$ edges from $G$ to obtain a graph in $\mathcal{F}$. A hereditary family is one closed under taking induced subgraphs, and it is proper if it is not the family of all graphs.

Conjecture on hereditary families close to $\mathcal{F}$. For each proper hereditary family $\mathcal{F}$ of graphs there are $C_{\mathcal{F}},d_{\mathcal{F}}>0$ such that if $0<\varepsilon<1/2$, every graph on $n$ vertices which is $\varepsilon^{d_{\mathcal{F}}}$-close to $\mathcal{F}$ contains an $\varepsilon$-homogeneous set of size at least $\varepsilon^{C_{\mathcal{F}}}n$.

This conjecture extends the polynomial Rödl phenomenon from hereditary families themselves to graphs that are polynomially close to such a family. The source calls it a natural conjecture for hereditary families and provides no resolution, so its general status remains open.