Glick's conjecture on the point of collapse of axis-aligned polygons

Let $P\subset\mathbb{P}^2$ be an axis-aligned $2n$-gon. Let $T$ denote the pentagram map, and let

$$p=T^{n-1}(P)$$

be its point of collapse. Let $\mathscr C(P)$ denote the center of mass of $P$. Glick's conjecture. One has

$$p=T^{n-1}(P)=\mathscr C(P).$$

Thus, the point of collapse is equal to the center of mass. Glick's observation arose from computer experimentation; the claim concerns the collapse described by the preceding theorem for generic axis-aligned polygons. Its resolution is not specified in the supplied text.