The short-edge crossing-number conjecture

Let $h(n)$ denote the maximum, over complete $n$-vertex simple topological graphs, of the minimum number of crossings of an edge in a suitable noncrossing matching construction. Short-edge crossing-number conjecture. For every integer $n\geq 2$,

$$h(n)=\Theta(n^{3/2}).$$

This would improve the currently established bound and sharpen the quantitative understanding of short edges in complete topological graphs; the source gives no resolution of the conjecture.