Existence of alternating quadrisecants for nontrivial knots

A quadrisecant of a knot is a straight line that intersects the knot four times. It is alternating when the four intersection points alternate between the north and south poles of the visual sphere, equivalently when their linear ordering along the quadrisecant alternates with their circular ordering along the knot. Alternating quadrisecant conjecture. Every nontrivial knot has an alternating quadrisecant. Such a quadrisecant would provide an alternate proof of the theorem that the second hull of a nontrivial knot is nonempty, because its middle segment lies in the second hull.