Existence of the Minkowski dimension of arboreal Galois images

Let $K$ be a number field, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a rational map of degree $d\geq2$ defined over $K$, and let $\alpha\in{\mathbb P}^1(K)$. Minkowski-dimension existence conjecture. The Minkowski dimension $\dim(G_{f,\alpha})$ exists. This conjecture asks whether the lower and upper Minkowski dimensions of every arboreal Galois image agree; the paper presents it as the first of several conjectures, while examples and calculations establish existence in a number of cases.