The Hamiltonian-path strengthening of Rafla's conjecture

Let $K_n$ be the complete graph, let $\mathcal{D}$ be a simple drawing of it on $n\ge 2$ vertices, and let $v$ and $w$ be distinct vertices of $\mathcal{D}$. A crossing-free Hamiltonian path is a path that visits every vertex exactly once, has no crossing edges, and whose end-vertices are $v$ and $w$. Hamiltonian-path strengthening. Every simple drawing $\mathcal{D}$ of $K_n$ on $n\ge 2$ vertices contains, for each pair of vertices $v \neq w$ in $\mathcal{D}$, a crossing-free Hamiltonian path with end-vertices $v$ and $w$.

The source states that this conjecture strengthens Rafla's Hamiltonian-cycle conjecture and that the implication between the two holds for the drawing classes considered there. It was confirmed for cylindrical and strongly c-monotone drawings, but the supplied text leaves the assertion open for arbitrary simple drawings.