The discrete Segre flattening conjecture for spherical polygons

Let $P$ be an embedded closed polygon in $S^2$ that bisects the area, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Discrete Segre conjecture. The polygon $P$ has at least $4$ flattenings. This is the discrete analogue of a smooth result attributed in the source to B. Segre and V. Arnold; the source presents the statement as a conjecture and seeks a specifically discrete proof.