Pre-compactness of pentagram-map iterates modulo affine transformations

Let $P\in B_{n,k}$, and write $P^{\ell}=\Delta^{\ell}(P)$ for $\ell=0,1,2,\ldots$. The sequence consists of $k$-birds in the space $B_{n,k}$.

Pre-compactness conjecture. The sequence $\{P^{\ell}\}$ is pre-compact modulo affine transformations. That is, this sequence has a convergent subsequence which converges to another element of $B_{n,k}$.

This conjecture concerns the long-term behavior of the generalized pentagram map. The supplied text gives no resolution, so its status remains open.