5 problems

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.

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…

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…

Henselianity conjecture. The valuation ring $\mathcal{O}$ is henselian.

Henselianity conjecture. The valuation $v$ is henselian.