Alon's asymptotic subgraph-count conjecture

Let $H$ be a fixed finite simple graph without isolated vertices. For a graph $G$, let $N(G,H)$ be the number of subgraphs of $G$ isomorphic to $H$, let

$$N(m,H)=\max\{N(G,H):e(G)=m\},$$

and define

$$\gamma(H)=\frac{|V(H)|+D(H)}{2},\qquad D(H)=\max_{S\subseteq V(H)}\bigl(|S|-|N_H(S)|\bigr).$$

Alon's asymptotic subgraph-count conjecture. There is a positive constant $b(H)$ such that

$$\lim_{m\to\infty}\frac{N(m,H)}{m^{\gamma(H)}}=b(H).$$

Alon proved the order of growth $N(m,H)=\Theta_H(m^{\gamma(H)})$ for every fixed $H$, and determined the exact leading constant when $D(H)=0$. The conjecture asks for the existence of the asymptotic leading constant in general.