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?