The conjecture on the -topology

Let $K$ be a large field, and let the $\mathscr{E}_K$-topology denote the topology introduced in the paper. A topology is gt-henselian when it is a gt-henselian topology on $K$.

The $\mathscr{E}_K$-topology conjecture. If $K$ is large, then the $\mathscr{E}_K$-topology is the intersection of all gt-henselian topologies on $K$. Consequently, the $\mathscr{E}_K$-topology is a field topology if and only if there is a coarsest gt-henselian topology on $K$.

For countable large fields this equivalence is known, while the conjecture asks for the general case.