Alon–Hefetz–Krivelevich–Tyomkyn sparse-edge decay 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

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

Sparse-edge decay conjecture. For all $k,\ell\in\mathbb{N}$ with

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

we have

$$\operatorname{ind}(k,\ell)=o_k(1).$$

This conjecture predicts that the asymptotic inducibility tends to zero when the induced edge count and its complementary count both grow faster than linearly in $k$. The supplied text does not state whether this conjecture has been resolved.