Alon–Hefetz–Krivelevich–Tyomkyn quadratic-range edge-statistics conjecture

For a graph $G$, let $X_{G,k}$ be the number of edges induced by a uniformly random $k$-vertex subset, and define

$$I(n,k,\ell):=\max\{\mathbb{P}[X_{G,k}=\ell]:v(G)=n\},$$

with

$$\operatorname{ind}(k,\ell):=\lim_{n\to\infty}I(n,k,\ell).$$

Quadratic-range edge-statistics conjecture. For all $k,\ell\in\mathbb{N}$ with

$$\min\left\{\ell,\binom{k}{2}-\ell\right\}=\Omega_k(k^2),$$

we have

$$\operatorname{ind}(k,\ell)=O_k(k^{-1/2}).$$

This is a quantitative strengthening in the dense interior range of the edge-count parameter, predicting a polynomial decay of the maximum asymptotic fraction of $k$-vertex subsets inducing exactly $\ell$ edges. The supplied text does not state whether this conjecture has been resolved.