Pentagram rigidity conjecture for convex Poncelet polygons

A convex Poncelet polygon is a polygon in the projective plane whose vertices lie on one conic and whose edges are tangent to another conic; convexity means convexity in an affine patch. Let $(n,k)$ be relatively prime integers with $n\geq 7$ and $k\in(2,n/2)$. For a convex $n$-gon $P_0$, define its $k$-diagonal orbit by $P_j=T_k^j(P_0)$, where $T_k$ maps an $n$-gon to the $n$-gon formed by the successive intersections of its $k$-diagonals. Pentagram Rigidity Conjecture. $P_0$ is convex Poncelet if and only if $P_j$ is convex for every $j\in\mathbb{Z}$.

This conjecture characterizes convex Poncelet polygons by the preservation of convexity along the entire deep-diagonal orbit. The forward implication is known because Poncelet polygons are projectively equivalent to their images under $T_k$; the converse is the conjectural rigidity statement, with this paper establishing a special case.