The plane-path conjecture for complete simple topological graphs

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. A plane path is a path whose edges are pairwise noncrossing. The length of a path is its number of edges.

Plane-path conjecture. There is an absolute constant $\varepsilon>0$ such that every complete $n$-vertex simple topological graph contains a plane path of length $n^{\varepsilon}$.

This conjecture asks for a polynomial-length plane path in every complete simple topological graph. The context notes that related results guarantee many pairwise disjoint edges, but the corresponding polynomial bound for plane paths remains open.