7 problems

Valuation conjecture. If $K$ is an unstable dp-finite field, then $K$ is valuation type.

Valuation conjecture. The group $I_K$ is an ideal in a valuation ring on $\mathbb{K}$; equivalently, $\operatorname{Stab}(I_K)$ is a valuation ring.

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

Valuation-type conjecture. If $\mathbb{K}$ is a dp-finite field, then $\mathbb{K}$ is stable or has valuation type.

Multi-valuation ideal conjecture. There is a small submodel $K \preceq \mathbb{K}$ for which the infinitesimals $I_K$ contain a nonzero multi-valuation ideal.

Henselianity conjecture. The valuation $v$ is henselian.

Let $K$ be a field, let $R\subseteq K$ be a good Bezout domain, and let $I_M$ denote the set of $M$-infinitesimals. The $M$-infinitesimals $I_M$ are an ideal in $R$. This would pro…