The convex-drawing extension conjecture for plane Hamiltonian cycles

Let $D$ be a convex drawing of the complete graph $K_n$, and let $C$ be a plane Hamiltonian cycle in $D$. A plane Hamiltonian subdrawing is a crossing-free subdrawing spanning all vertices.

Convex-drawing extension conjecture. Every plane Hamiltonian cycle $C$ in $D$ can be extended to a plane Hamiltonian subdrawing on $2n-3$ edges.

The conjecture asks whether every prescribed plane Hamiltonian cycle in a convex drawing can be enlarged to the target-size subdrawing. The paper reports that this holds computationally for convex drawings with $n\leq 10$, while the analogous assertion fails for general simple drawings.