The edge-statistics conjecture for $\mathrm{\ell }\ll k^{6\mathrm{/}5}$

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