Rational closures of group rings of left-ordered groups

Suppose $K$ is a division ring, and $G$ is a left-ordered group such that for any Dedekind cut $\varepsilon$ of the linearly ordered set $(G,\le)$ the group $S=\{g\in G\mid g\varepsilon=\varepsilon\}$ is such that $KS$ is a right Ore domain and the group $H=\{g\in G\mid gP(G)g^{-1}=P(G)\}$ is cofinal in $G$. Then the group ring $KG$ can be embedded in a division ring having a valuation in the sense of Mathiak with values in $G$. If $G$ is the group of a trifolium, this construction leads to an example of a chain domain with a prime, but not completely prime, ideal.