Classification conjecture for dp-minimal expansions of the integers

Let $\mathcal{Z}$ be a dp-minimal proper expansion of $(\mathbb{Z},+,0,1)$. A classification conjecture for dp-minimal expansions of the integers. Exactly one of the following holds:

1. $\mathcal{Z}$ is interdefinable with $(\mathbb{Z},+,0,1,<)$.
2. There is $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the cyclic order $C_{\alpha}$ induced by $n\mapsto n\alpha+\mathbb{Z}$ is definable in $\mathcal{Z}$.
3. There is a generalized valuation $v$ such that the associated relation $\preceq_v$ is definable in $\mathcal{Z}$.

This conjecture aims to classify all dp-minimal proper expansions of the additive structure of the integers by separating the ordered, irrational cyclic-order, and generalized-valuation cases. The surrounding discussion notes that many proper dp-minimal expansions are known, while a complete classification remains open.