The saturated large-field conjecture for the $\mathscr{E}_K$-topology

Let $K$ be a large, $\lambda$-saturated field for a sufficiently large cardinal $\lambda$. For a henselian local domain $R\subseteq K$ with fraction field $K$, let the $R$-adic topology be the topology whose basic sets are $aR+b$ with $a\in K^\times$ and $b\in K$.

Saturated large-field conjecture. The $\mathscr{E}_K$-topology is the intersection of all $R$-adic topologies as $R$ ranges over henselian local domains $R\subseteq K$ with $\operatorname{Frac}(R)=K$. Moreover, it is a field topology if and only if there is a henselian local domain $R\subseteq K$ with $\operatorname{Frac}(R)=K$ such that, for every other such henselian local domain $S\subseteq K$, one has $aS\subseteq R$ for some $a\in K^\times$.

The paper relates this claim to the description of $\mathscr{E}_K$-open sets in highly saturated large fields; its resolution is not supplied here.