In this question, David Loeffler asked if one can show that the (local) Artin map $$K^\times \to G_K^{ab}$$ does not have a lift to $G_K$. Probably this wouldn't be canonical, but I can't show that such a lift doesn't exist.

So does there exist such a lift?

In a comment to this question, David Loeffler asked if one can show that the (local) Artin map $$K^\times \to G_K^{ab}$$ does not have a lift to $G_K$. Probably this wouldn't be canonical, but I can't show that such a lift doesn't exist.

So does there exist such a lift?