Affine dynamics of the polygon iteration on ordinary polygon space

Let $n \geq 5$, let $H_{\lambda}$ be the polygon iteration on the space $\mathcal{P}_n=(\mathbb{R}P^2)^n$ of $n$-gons, and let $\lambda_0=2\cos(\pi/n)$. Affine dynamics conjecture. If $\lambda \in (0,\lambda_0)$, any $n$-gon collapses to a point under iteration of $H_{\lambda}$. If $\lambda=\lambda_0$, then for any $n$-gon $P$, the even iterates $H_{\lambda}^{2k}(P)$ converge, without rescaling or other normalization, to a regular polygon and the odd iterates $H_{\lambda}^{2k+1}(P)$ converge to another regular polygon; the vertices of these two polygons form a regular $2n$-gon. If $\lambda \in (\lambda_0,\infty)$, then every convex $n$-gon degenerates under iteration, with all its vertices approaching a straight line. This conjecture describes the parameter-dependent behavior of the iteration on the unquotiented polygon space, including collapse, alternating regular-polygon limits, and degeneration to a line. The supplied text says these predictions are guided by computer experiments and gives no resolution.