The $2n-3$-edge plane Hamiltonian subdrawing 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 subdrawing is a plane subdrawing containing all $n$ vertices and a Hamiltonian cycle.

$2n-3$-edge conjecture. Every simple drawing of $K_n$ with $n\geq 3$ contains a plane Hamiltonian subdrawing on $2n-3$ edges.

This is the second strengthening of Rafla's conjecture stated in the conclusion. The source presents it as conjectural for general simple drawings, with computational evidence for a related weakening.