The topology conjecture for proper dp-minimal expansions of $(\mathbb{Z},+)$

Let ${\mathscr{Z}}$ be an expansion of $(\mathbb{Z},+)$, and say that it is proper if it is not interdefinable with $(\mathbb{Z},+)$. Say that ${\mathscr{Z}}$ eliminates $\exists^\infty$ if, uniformly in parameters, it cannot define finite sets of arbitrarily large cardinality. A group topology on $(\mathbb{Z},+)$ is non-discrete if the singleton topology is not the topology in question.

Topology conjecture. Any proper dp-minimal expansion of $(\mathbb{Z},+)$ which eliminates $\exists^\infty$ defines a non-discrete group topology on $(\mathbb{Z},+)$.

The conjecture proposes that all proper dp-minimal expansions with this finiteness property arise from a non-discrete group topology, complementing the known result that expansions failing to eliminate $\exists^\infty$ define the order $<$.