Gerritzen–van der Put's position conjecture for split degenerate superelliptic curves

Let $S \subset \mathbb{P}_K^1$ be the set of fixed points of generators of a $p$-Whittaker group $\Gamma_0$, and let $\mathcal{B} \subset \mathbb{P}_K^1$ be its image under reduction modulo $\Gamma_0$, namely the set of branch points of the resulting superelliptic curve. For a finite subset $A \subset \mathbb{P}_K^1$ of cardinality at least $3$, its position is the combinatorial data of the tree $T(A)$ together with the map from $A$ to the vertices of $T(A)$ determined by the components containing the reductions of its points; two equally large subsets have the same position when a bijection between them is compatible with an isomorphism of their trees.

Gerritzen–van der Put's conjecture. With this setup, the $(2g+2)$-element subsets $S \subset \mathbb{P}_K^1$ and $\mathcal{B} \subset \mathbb{P}_K^1$ have the same position.

The conjecture proposes that the configuration of the fixed points and the configuration of the branch points agree combinatorially. The paper disproves it by a counterexample.