Let $K$ be a finite extension of the $p$-adic numbers. $G_K$ be its absolute Galois group and $I_K$ the inertia subgroup. Are finite index subgroups of $I_K$ closed in its profinite topology?

By a result of Nikolov/Segal it suffices to show, that inertia is topologically finitely generated.

What has been proved in the direction of this or the analogous question on wild inertia?

Do the results of Jannsen/Koch/Wingberg on the finite generator rank of the local Galois group help? I couldn't find anything on this and considering its importance, it surely would be mentioned somewhere, if it were always known. So maybe someone has some conditional results?

arefinite-index subgroups of $I_K$ that are not closed (for reasons indicated in my preceding comment, since a countably infinite direct product of $\mathbf{F}_p$'s has only countably many closed subgroups of finite index but has uncountable dimension over $\mathbf{F}_p$ and hence has uncountably many finite-index subgroups, so "most" of them are not closed). $\endgroup$ – user29283 Apr 24 '13 at 16:15