Aichholzer–Orthaber–Vogtenhuber's plane Hamiltonian-connectedness conjecture

Let $K_n$ be the complete graph, and let a simple drawing be a drawing in which vertices are distinct points, edges are simple curves joining their endpoints and avoiding other vertices, and every two edges meet at most once, either at a common endpoint or at a proper crossing. A plane Hamiltonian path is a Hamiltonian path whose edges do not cross.

Aichholzer–Orthaber–Vogtenhuber's conjecture. For each pair of vertices $s,t$ in a simple drawing of $K_n$, there exists a plane Hamiltonian path from $s$ to $t$.

This is a strengthening of Rafla's conjecture, requiring Hamiltonian paths with every prescribed pair of endpoints. The source introduces it as a variant of Rafla's conjecture; its general status remains open.