Concentrate quadratically many edges in a short crossing-profile interval

Determine whether, for every n, there exists a rectilinear drawing of K_n and positive integers m and ell with ell=o(n^2) such that the sum of the crossing-profile entries from m through m+ell is Omega(n^2).

Background

Beyond individual crossing-profile entries, the paper asks whether a substantial fraction of all edges can be concentrated within a short range of crossing counts. The proposed condition requires an interval whose length is subquadratic in n to contain a quadratic number of edges. The authors contrast this unresolved question with the currently known convex-position construction, which only gives an upper-scale concentration of order Omega(n\sqrt{ell}) for an interval of length ell.

References

For example, is it true that for every $n$ there exists a rectilinear drawing $\mathcal D$ and two positive integers $m$ and $\ell$ such that $\ell=o(n2)$ and $\sum_{k=m}{m+\ell}S_k(\mathcal D)=\Omega(n2)$?

— On the crossing profile of rectilinear drawings of $K_n$  (2501.04980 - Chen et al., 9 Jan 2025) in Section 7.1, "Regarding overline{max} e_k(K_n"