Derivations and gt-henselian field topologies

Suppose that $K$ is a characteristic zero field with infinite transcendence degree over its prime subfield. We show that if there is a gt-henselian topology on $K$ then there are $2^{2^{\mathrm{\mid }K\mathrm{\mid }}}$ pairwise incomparable gt-henselian topologies on $K$. It follows by applying a recent theorem of Will Johnson that if $K$ is large and countable then there are $2^{2^{{\mathrm{\aleph }}_0}}$ pairwise incomparable gt-henselian topologies on $K$. We also formulate several conjectures concerning gt-henselian field topologies and their relationship with the étale-open topology.