At UC BERKLEY in the 1960's there were protests because, while the hired Communists, the speeches at rally COULD be censored. What fifty years of progress at a progressive institution brings will bring when Progressives are in charge, eh?
In my experience, liberals have ALWAYS argued to limit conservative free speech. The effects are obvious. Read The Spiral of Silence : Public Opinion – Our Social Skin by Elisabeth Noelle-Neumann.
Public opinion theory based on studying the French Revolution. Social pressure stifles politically incorrect speech. This "silence" is self-reinforcing, as people that would feel comfortable to speak out if they were supported by others do not hear others expressing their own dissent opinions. The dominate opinion becomes uncritical and more oppressive. Why do you think Trump polled lower than his actual electoral outcome. Trump was politically incorrect and people were not honest to pollsters due to perceived social pressure. It's mirror image is "The Bradley Effect". https://en.wikipedia.org/wiki/Bradley_effect
https://en.wikipedia.org/wiki/Spiral_of_silence