The vanishing-density conjecture for pentagram-map dynamical domains

Let $p$ be a prime, and let $f$ denote the pentagram map on the moduli space $\mathcal{T}_n$ of twisted $n$-gons over $\overline{\mathbb{F}}_p$. Let $\operatorname{DynDom}(f)$ be the set of points whose forward orbits avoid the indeterminacy locus of every iterate.

Vanishing-density conjecture. The dynamical domain has asymptotic density zero over the tower of finite fields:

$$\lim_{r\to\infty}\frac{\#\operatorname{DynDom}(f)\cap\mathcal{T}_n(\mathbb{F}_{p^r})}{\#\mathcal{T}_n(\mathbb{F}_{p^r})}=0.$$

This conjecture predicts that almost every orbit of the pentagram map over finite fields eventually reaches the degeneracy locus. It is presented in the paper as the proposed counterexample to the general large-dynamical-domain conjecture, motivated by computer experiments and the paper's main theorem.