The hyperplane-intersection flattening conjecture for strictly convex projective polygons

Let $P$ be a strictly convex polygon in $\mathbb{RP}^d$. Suppose that $P$ intersects a hyperplane with multiplicity $k$, where $k\geq d+2$. Call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Hyperplane-intersection flattening conjecture. The polygon $P$ has at least $k$ flattenings. The source notes that this statement is stronger than the preceding theorem, but gives no resolution or further evidence beyond presenting it among three discrete conjectures.