Asymptotic reduction conjecture for graph blow-ups

Let $H$ be a graph on $m$ edges, let $k$ be a positive integer, let $\delta(H)$ be the minimum degree of $H$, and let $\mathcal{G}_C$ be the graph class defined in the paper. Write $H\{k\}$ for the $k$-subdivision (edge-blow-up) of $H$, and let $\beta(H,k)$ be the associated optimization parameter. Graph-blow-up reduction conjecture. If $k\delta(H)\geq 2$, then

$$\operatorname{\mathbf{N}}_{\mathcal{G}_C}(n,H\{k\})={\beta(H,k)\over(k!)^m}\cdot n^{km}+o(n^{km}).$$

This conjecture would establish the expected asymptotic count in the relevant graph class under the stated minimum-degree condition. It is presented as a generalization of the paper's reduction lemmas and remains open in the source.