Improve the upper bound for the maximum k-crossing index

Prove that, for every n and k, the maximum number of edges involved in exactly k crossings in a rectilinear drawing of K_n is asymptotically smaller than n\sqrt{k}, namely establish overline{max} e_k(K_n)=o(n\sqrt{k}).

Background

The paper establishes the bounds Omega(n) <= overline{max} e_k(K_n) <= O(n\sqrt{k}) over the parameter range in which an edge with k crossings can occur. The authors state that they believe the upper bound can be improved and formulate a conjecture asserting a little-o improvement over n\sqrt{k}. They note that resolving the conjecture would be meaningful even for fixed k or for k of order n^2.