The characterization of minimal distal valuation expansions of the integers

Let $(B_i)_{i<\omega}$ be a strictly descending chain of subgroups of $\mathbb{Z}$ with $B_0=\mathbb{Z}$, and let $v:\mathbb{Z}\to\omega\cup\{\infty\}$ be the associated valuation. A valuation expansion $(\mathbb{Z},+,0,1,v)$ is distal when it has the model-theoretic distality property.

Characterization conjecture. Let $(\mathbb{Z},+,0,1,v)$ be distal. Then the following are equivalent:

1. $(\mathbb{Z},+,0,1,v)$ is a minimal expansion of $(\mathbb{Z},+,0,1)$.
2. There is a prime $p$ such that $|B_i/B_{i+1}|=p$ for almost all $i<\omega$.
3. $v$ is interdefinable with a $p$-adic valuation for some prime $p$.
4. The $(\mathbb{Z},+,v)$-induced structure on the value set of $v'$ is o-minimal for every $(\mathbb{Z},+,v)$-definable valuation $v'$.

The paper explains that arbitrary valuation expansions can be dp-minimal and that distal examples need not be minimal. The conjecture proposes that, among distal valuation expansions, minimality is equivalent to eventual prime-sized successive quotients, interdefinability with a $p$-adic valuation, and the stated o-minimality condition.