Halupczok's tree-level conjecture for definable p-adic sets

Let $X \subset \mathbb{Q}_p^n$ be a definable set in the language of fields. For each $\lambda \geq 0$, let $X_\lambda$ be the image of $X \cap \mathbb{Z}_p^n$ in $(\mathbb{Z}/p^\lambda\mathbb{Z})^n$, and let

$$\operatorname{T}(X):=\bigcup_{\lambda\geq 0}X_\lambda$$

with its natural tree structure. The dimension of $X$ is the algebraic dimension of its Zariski closure in the algebraic closure $\widetilde{\mathbb{Q}}_p^n$. Halupczok's tree-level conjecture. The tree $\operatorname{T}(X)$ is a tree of level $\dim X$. This conjecturally gives a purely combinatorial description of trees arising from definable sets and would refine Denef's rationality result for the associated Poincaré series. The paper proves the claim in several special cases, but its general status is not resolved.