In a comment to this [question](https://mathoverflow.net/questions/429009/non-existence-of-higher-artin-map), 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?