4 problems

Figure-eight region-count conjecture. The constraint reduces the number of crossings by exactly $2$ for each figure $8$, and consequently

Affine-family intersection conjecture. If $S$ is a shape belonging to the affine family, then equality can always be achieved in the displayed bound.

Conjugation-free extension conjecture. Then

Independence conjecture. The property of having a conjugation-free geometric presentation is independent of the choices of $O$, $cell$ and $u\incell$.