The crossing-degree conjecture for complete simple topological graphs

Let $h=h(n)$ be the smallest integer such that every complete $n$-vertex simple topological graph contains an edge crossing at most $h$ other edges. A simple topological graph is a graph drawn in the plane so that vertices are distinct points, edges are simple arcs joining their endpoints, no edge contains a vertex other than its endpoints, and every pair of edges intersects at most once, either at a common endpoint or at a proper crossing.

Crossing-degree conjecture. There is an absolute constant $\varepsilon>0$ such that

$$h(n)\leq n^{2-\varepsilon}.$$

The source records the lower bound $h(n)=\Omega(n^{3/2})$ and upper bounds of order $n^2/(\log n)^{1/2-o(1)}$. The conjecture seeks a polynomial improvement over the quadratic upper bound and remains open.