if I remember correctly from my con law class in high school, the first amendment indeed does not cover lies. all they have to do is claim that he is lying or spreading false truths and they can censor whatever they want. it doesn't matter if the person they are censoring believes they are telling the truth, let alone that what they're saying is actually the truth. if they can convince a judge that you are lying, they have plausible deniability to revoke your first amendment rights.

Actually, I believe that is incorrect. In 2012, United States vs Alvarez, the Supreme Court struck down the so-called Stolen Valor act (which made it illegal to lie about having received military medals), saying that it violated free speech provisions.

Can't recall if they ruled on anything else in the past decade that overturned that, but going off Alvarez you can lie and have it covered under free speech.