Super-linear edge-statistics conjecture

Let $\operatorname{ind}(k,\ell)$ denote the limiting maximum proportion of $k$-vertex subsets inducing exactly $\ell$ edges. Super-linear edge-statistics conjecture. For all pairs $(k,\ell)$ satisfying

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

we have

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

The paper proves the analogous assertion when both edge counts are quadratic in $k$ and proposes replacing “quadratic” by “super-linear.” This strengthening remains open.