Trying to solve bad speech by censorship will never work, and there has never been an authoritarian movement that has been silenced by censorship. It do not work. At best it does nothing, and at worst it increase their support and reduce any internal resistance. The purpose of human rights is to create common shared values between people from all political spectrum and create some internal resistance.
It is fairly common belief that lies and propaganda is stronger than truth. However rather than see it as a fundamental part of nature, I would propose an alternative theory. Lies and propaganda is a symptom when social trust in society start to break down. You could ban Fox news or any other right-leaning media and there is little to no evidence that society would be any different.
The idea of free speech absolutism is an concept that people build to attack free speech. Free speech is about valuing and believing that people should be free to hold and express their opinions and beliefs without fear of retaliation. Fear can be created by law, by mobs or by those who hold some form of power like employers, but regardless of method the result is the same. Society need to value the idea of free speech. Absolutism has nothing to do with that. The idea that people should not fear the government for beliefs they hold, but should fear their employer, is inconsistent with free speech as a human right.
If you are looking for something more concrete, I would point towards research that that looks at social trust and its roles in conflicts. There exist a fair amount of research on this topic, some which is left politics and others which is right politics. One major finding are the importance of shared values. What kind of values those are is less important than that they are shared. If they aren't shared, then the next most important part is that they aren't shoved into the face of people who do not share them. Trying to stamp out opposing values, especially in a public and diffused way, has a long history of creating violence, fear and mistrust.