The discrete Möbius flattening conjecture for non-contractible polygons

Let $P$ be an embedded non-contractible closed polygon in $\mathbb{RP}^2$, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. Discrete Möbius conjecture. The polygon $P$ has at least $3$ flattenings. In the smooth case this is the celebrated Möbius theorem, where flattening means inflection; the discrete statement is proposed as its analogue, and the source calls for a specifically discrete proof.