All-pairs plane Hamiltonian path conjecture for simple drawings

Let $K_n$ be drawn simply in the plane, and let $a,b$ be two vertices. A path is plane if no two of its edges cross. All-pairs Hamiltonian-path conjecture. For every simple drawing of $K_n$ and every choice of two vertices $a,b$, there is a plane Hamiltonian path from $a$ to $b$. The conjecture is verified for $n\leq 10$ in simple drawings and for $n\leq 13$ in generalized twisted drawings, and is proved for several subclasses; the general case remains open.