The $p$-adic completion conjecture for dp-minimal expansions

Let ${\mathscr{Z}}$ be a dp-minimal expansion of $(\mathbb{Z},+,\operatorname{Val}_p)$. Let ${\mathscr{Z}}^{\square}$ denote its completion and ${\mathscr{Z}}^{\mathrm{Sh}}$ its Shelah expansion.

$p$-adic completion conjecture. The structure induced on $\mathbb{Z}$ by ${\mathscr{Z}}^{\square}$ is interdefinable with ${\mathscr{Z}}^{\mathrm{Sh}}$, and every ${\mathscr{Z}}^{\mathrm{Sh}}$-definable subset of $\mathbb{Z}^n$ has the form $X\cap Y$, where $X$ is a ${\mathscr{Z}}^{\square}$-definable subset of $\mathbb{Z}_p^n$ and $Y$ is a $(\mathbb{Z},+)$-definable subset of $\mathbb{Z}^n$.

This predicts a precise relationship between the completion and Shelah expansion of any dp-minimal expansion defining the $p$-adic valuation. The assertion is presented as an open problem in the paper.