The polynomial noncrossing-path conjecture

A complete $n$-vertex simple topological graph is a drawing of the complete graph on $n$ vertices in which vertices are points and edges are simple curves, with no edge passing through a nonincident vertex and with any two edges meeting at most once. A noncrossing path is a path whose edges do not cross. Polynomial noncrossing-path conjecture. There is an absolute constant $\varepsilon>0$ such that every complete $n$-vertex simple topological graph contains a noncrossing path on $n^{\varepsilon}$ vertices. The conjecture would improve the known lower bound of order $\frac{\log n}{\log\log n}$ for noncrossing paths; the source states no resolution.