Positive fixed-point proportion dichotomy for polynomial iterated Galois groups

Let $K$ be a field and let $f\in K[x]$ be a polynomial of degree $d\geq 2$. Assume either that $\operatorname{char}(K)=0$ or that $\operatorname{char}(K)$ does not divide the local degree of any critical point of $f$. Let $K^{\mathrm{sep}}$ be a separable closure of $K$, let $t$ be transcendental over $K$, and let $G_\infty(K^{\mathrm{sep}},f,t)$ be the geometric iterated Galois group. A polynomial has a euclidean orbifold when its associated orbifold is euclidean; the type $(\infty,\infty)$ is excluded below. Positive fixed-point proportion dichotomy conjecture. Either $f$ has a euclidean orbifold not of type $(\infty,\infty)$ and

$$\mathrm{FPP}(G_\infty(K^{\mathrm{sep}}, f, t))>0,$$

or

$$\mathrm{FPP}(G_\infty(K^{\mathrm{sep}}, f, t))=0$$

otherwise. The result would further classify polynomial iterated Galois groups with positive fixed-point proportion. The surrounding discussion derives the expected dichotomy from known martingale and mixing results, but the conjectural classification itself remains open in the supplied text.