I do agree with Jack Awful, in that conservatives only care about freedom of speech because they're the ones being censored right now and liberals are only against it because they aren't being censored. Just look at mid 20th century america if you need any more evidence of that.
"Conservative" is a very broad label that describes a massive group, and the composition of such groups inevitably changes with time. There is a very large mass of people who are fundamentally fearful, irrational, and uncritical who become authoritarians, and thus are hateful, tribalistic, hypocritical, and unempathetic. And because it's far easier to change
what you think than to change
how you think, they can be either Left or Right, and this identification can change during their lifetime. Because they can be bullied and manipulated into compliance, and are bullies themselves, they can form a critical mass of such people with like beliefs that grows naturally until it comprises most of the susceptible population, and becomes the dominant political force.
But since they are such utter assholes, their kids inevitably hate them and rebel and run in the opposite direction, bringing their reign to an end and instituting a new one on the other side of the political spectrum. But since their parents never taught them how to think, because they themselves didn't know how, the next generation becomes just as bad as the old one. And the cycle repeats. This is why the pendulum of American politics swings on a generational timeline. In the mid 20th century during the Red Scare, anticommunist conservatives had most of the authoritarians. Today, it's the Social Justice movement that has most of the authoritarians. Whichever side has the critical mass of authoritarians, will be the side that wants to censor people. Count on it.
Authoritarians have poor self-monitoring ability, they aren't any better at looking at their own thoughts critically than they are looking at anyone else's. And because they therefore can't control themselves, they must control others. They
have to shut down opposing ideas by force, because they don't have the ability to defend against them logically. Deep down, they know that, and it terrifies them. And they assume everyone else is just as gullible, so they
must protect society from evil ideas. If conservatives take over and
immediately start censoring people, it will only be because they are so frightened by how close we've come to the absolute abyss that they have decided that
anything is better than that. And this will be historically unprecedented, and horrifying. But if things follow the historical pattern, we'll just get another normal pendulum swing.