Pikhurko–Sousa conjecture on asymptotic extremal decompositions

Let $H$ be a graph with chromatic number at least $3$. For an $n$-vertex graph $G$, let $\phi(G,H)$ be the minimum number of parts in an edge partition into copies of $H$ and single edges, and define $\phi(n,H):=\max_{G\in\mathcal{G}_n}\phi(G,H)$. Let $\operatorname{ex}(n,H)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $H$.

Pikhurko–Sousa conjecture. There is an integer $n_0=n_0(H)$ such that

$$\phi(n,H)=\operatorname{ex}(n,H)$$

for every $n\geq n_0$.

Pikhurko and Sousa had previously established the asymptotic relation $\phi(n,H)=\operatorname{ex}(n,H)+o(n^2)$ for each fixed graph $H$ with chromatic number at least $3$; the conjecture asks for eventual exact equality.