Stabilizer-ring conjecture for saturated unstable dp-finite fields

Let $\mathbb{K}$ be a saturated unstable dp-finite field. A subgroup $J \subseteq (\mathbb{K},+)$ is bounded in the sense used by the source, and its stabilizer ring is

$$R_J = \{x \in \mathbb{K}: xJ \subseteq J\}.$$

Stabilizer-ring conjecture. There is a non-zero bounded subgroup $J \subseteq (\mathbb{K},+)$ for which $R_J$ is a multi-valuation ring on $\mathbb{K}$.

The source states this as an equivalent formulation of the valuation-type conjecture. The supplied text does not resolve it.