Alon–Hefetz–Krivelevich–Tyomkyn square-root anti-concentration conjecture for graph edge statistics

Let $G$ be an $n$-vertex graph. For $0\leq k\leq n$, choose a uniformly random $k$-vertex subset $A\subseteq V(G)$ and let $X_{G,k}:=e(G[A])$ be the number of edges induced by $A$. Suppose $k\to\infty$ and $n/k\to\infty$, and let $\ell$ satisfy

$$\ell=\Omega(k^{2})\quad\text{and}\quad \binom{k}{2}-\ell=\Omega(k^{2}).$$

Alon–Hefetz–Krivelevich–Tyomkyn's conjecture. The edge statistic satisfies

$$\Pr(X_{G,k}=\ell)=O(1/\sqrt{k}).$$

This predicts Gaussian-scale anti-concentration for induced edge counts when both the edge count and its complement are of quadratic order. The source presents it as an open conjecture; no resolution is given here.