Alon–Hefetz–Krivelevich–Tyomkyn Poisson 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 $n$ grows sufficiently rapidly in terms of $k$. For every integer $\ell$ with

$$0<\ell<\binom{k}{2},$$

Alon–Hefetz–Krivelevich–Tyomkyn's conjecture. One has

$$\Pr(X_{G,k}=\ell)\leq 1/e+o(1).$$

This conjecture seeks a Poisson-type universal upper bound, analogous to the maximum point mass of a Poisson random variable with mean one. The source gives no resolution, so the conjecture remains open.