The sublinear upper-bound conjecture for the crossing profile of complete graphs

For a rectilinear drawing of the complete graph $K_n$, let $e_k(K_n)$ denote the number of edges crossed exactly $k$ times, and let

$$\overline\max\ e_k(K_n)$$

denote the maximum of $e_k(K_n)$ over all rectilinear drawings of $K_n$. Sublinear crossing-profile conjecture. For every $n\geq 1$ and $k\geq 1$,

$$\overline\max\ e_k(K_n)=o(n\sqrt{k}).$$

More precisely, for every $\epsilon>0$ there exists a constant $N$ such that if $n$ and $k$ are larger than $N$, then $\overline\max\ e_k(K_n)/(n\sqrt{k})<\epsilon$. The paper establishes the bounds $\Omega(n)\leq\overline\max\ e_k(K_n)\leq O(n\sqrt{k})$ in the stated range, and this conjecture proposes that the upper bound is asymptotically non-sharp.