Östlund's conjecture on simplifying generic circle immersions

A knot projection is the image of a generic immersion of a circle into the $2$-sphere, and deformations of types $1$, $2$, and $3$ are the local replacements analogous to Reidemeister moves. In particular, consider generic immersions

$$S^{1}\to\mathbb{R}^{2}$$

and embeddings of the circle. Östlund's conjecture. Deformations of types $1$ and $3$ are sufficient to obtain a homotopy from any generic immersion $S^{1}\to\mathbb{R}^{2}$ to an embedding. The conjecture was disproved by Hagge and Yazinski, who exhibited a first counterexample with $16$ double points; consequently, deformations of type $2$ can be necessary.