The definable-basis criterion for the $\mathscr{E}_K$-topology

Let $K$ be a field with a gt-henselian topology, and suppose that this topology has a basis consisting of definable subsets of $K$. Let the $\mathscr{E}_K$-topology be the topology defined in the paper.

Definable-basis conjecture. If $K$ admits a gt-henselian topology with a basis consisting of definable sets, then the $\mathscr{E}_K$-topology is a field topology.

The preceding discussion shows that the converse holds when the $\mathscr{E}_K$-topology is a field topology, with an existentially definable basis. The asserted converse implication is open in the supplied text.