  Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leq
o_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertex
subsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Together
with a recent result of Kwan, Sudakov, and Tran, this settles a conjecture of
Alon, Hefetz, Krivelevich, and Tyomkyn.
