I don't think "false information" is the kicker here. Taking a cursory glance at Breitbart and Fox News, there isn't that much that is patently false. Furthermore, my observations of how propaganda works in general is that it doesn't usually rely information that is technically false. It relies on selective distribution of information, editorial spin, and other more subtle methods.

This makes "enforcement" basically impossible because you can't have a news outlet without editorial decisions. A better way to go about it would be to try incentivize media outlets to make a good faith effort at "both sides" journalism, but members of both political sides have been attacking the media for doing just that since the 2016 election season.