The solution of driving cultural change to solve the problem of believing so-called fake news will run into the same problems as highlighted in this article (the school girl example in the starting is instructive).
One of things the article argues is that media literacy programs to solve for the fake news problem will backfire. They will backfire, it claims, because of different embedded beliefs. There are of course many underlying reasons for these different beliefs but chief among them, I suspect, will be different cultural upbringing. I would argue that cultural diversity is only going to increase (globalization, immigration, Internet, etc.). Therefore, in such a scenario bringing about cultural change to solve this problem is a non-starter of a solution.
Instead one should take as given all the points highlighted in this article leading up to this solution, and assume that in multi-cultural societies this problem is only going to deepen further. How then can we solve the problem of deliberate intentions to misinform?
It appears to be an age old problem of communication — how do you communicate with someone who follows a different belief system than yours? And to make matters, the person has different standards of trust than you do in different sources of news/information.
A large part of the solution might lie with institutions (or media companies) that want their authority and credibility to be universally accepted. To illustrate my suggestion let me phrase a different question which, instead of abstracting from the context, takes a specific instance. If NYTimes needs to establish its credibility within the alt-right sections of the society what should it do differently?