I think what they mean by truths becoming false is rather that all certainties become uncertainties. Delving too deep in postmodernism is like taking all of your legos and melting them.

If I want to have a conversation about free speech, and a postmodernist begins questioning if we even actually have free will, or if it's even valid to discuss morality if it's entirely possible we exist in a simulation...you haven't actually advanced anything. You've just made soup. Postmodernism is a tool for turning building blocks into soup, and that is more often than not extremely counterproductive. Though it is, to some degree, necessary.

I think the fundamental problem is that aggressive postmodernism will often disregard presuppositions with absolutely no interest in understanding the utility/value of the presupposition.