Projective equivalence conjecture for generalized pentagram maps

Let $P$ be a $(k+3)$-gon in $\mathbb{RP}^k$, where $k\geq 2$, and let $J=(1,\ldots,1)$. For even $k$, let $I=(1,\ldots,1,2,1,\ldots,1)$ with the same number of $1$'s before and after the $2$; for odd $k$, let $I=(1,1,\ldots,1,2,1,\ldots,1)$ with one more $1$ before the $2$ than after it. Projective equivalence conjecture. The polygons $P$ and $T_{I,J}(P)$ are projectively equivalent. Computer-based experiments suggest that this extends the known projective invariance of pentagons in $\mathbb{RP}^2$ and hexagons in $\mathbb{RP}^3$ under suitable generalized pentagram maps; the claim concerns projective self-duality phenomena for polygons in arbitrary higher dimensions and remains unproved in the supplied text.