Alon's star-forest extremal-graph conjecture

Let $H$ be a star forest, meaning a vertex-disjoint union of stars. For a graph $G$, let $N(G,H)$ be the number of subgraphs of $G$ isomorphic to $H$, and let

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

Alon's star-forest extremal-graph conjecture. For every integer $m>0$, or at least for all sufficiently large $m$, there exists a star forest $G_m$ with $e(G_m)=m$ such that

$$N(m,H)=N(G_m,H).$$

The conjecture describes the form of an extremal graph for counting a fixed star forest with a prescribed number of edges. The supplied text gives no resolution evidence, so whether the assertion holds for every $m$ or only eventually remains open.