Quantitative restricted partition conjecture for graphs

For $\varepsilon>0$, a graph $G$ is $(N,\varepsilon)$-restricted if its vertex set can be partitioned into at most $N$ subsets that are $\varepsilon$-restricted in $G$, where a subset is $\varepsilon$-restricted if one of its induced subgraph or complement has maximum degree at most $\varepsilon$ times its size. Let $\operatorname{ind}_H(G)$ denote the number of copies of a graph $H$ in $G$. Quantitative restricted partition conjecture. For every $\varepsilon>0$ and every graph $H$, there exist $N=N(H,\varepsilon)>0$ and $\kappa=\kappa(H,\varepsilon)>0$ such that every graph $G$ satisfying

$$\operatorname{ind}_H(G)\leq\kappa\abs{G}^{\abs H}$$

is $(N,\varepsilon)$-restricted. This would unify Nikiforov's quantitative theorem with the qualitative restricted-partition theorem and substantially strengthen the corresponding weakly restricted result.