Essential quadrisecants conjecture for the figure-8 and $5_2$ knots

A quadrisecant is a line meeting a knot in four points. A quadrisecant is essential when the secants joining consecutive points along the knot are essential according to the arcs between their endpoints. A flipped quadrisecant and a simple quadrisecant are the corresponding geometric types of quadrisecant described in the source. The figure-8 knot is the knot $4_1$, and $5_2$ denotes the standard knot with crossing number five.

Essential quadrisecants conjecture. The figure-8 knot has at least one essential flipped quadrisecant, and the $5_2$ knot has at least one essential simple quadrisecant.

This conjecture is motivated by examples found for low-crossing knots and is stated for knots of finite total curvature. The source gives no resolution.