Erdős Problem #59 — Counting H-free graphs at the extremal scale

Let $H$ be a fixed graph containing a cycle, and let $\operatorname{ex}(n,H)$ be the maximum number of edges in an $H$-free graph on $n$ vertices. Is the number of labelled $H$-free graphs on $n$ vertices $2^{(1+o(1))\operatorname{ex}(n,H)}$?