It has not been proven. Even if you apply a name to it, that doesn't prove persuasion doesn't work. If naming something proved it were true, then my naming the marketplace of ideas philosophy would have proven it's true.
That's absolutely something each speaker needs to deal with in their speech if they want to be persuasive. All that does it make it difficult. If changing minds were easy, everyone would change their mind about everything all the time.
A college debate class confronts these issues head-on. In terms of proof, you can "prove" to you that persuasion of some people is in fact possible even when they have deep bias.
Have you never changed anyone's mind about anything which they were predisposed to be biased to believe prior to you trying to persuade them? I mean, even when a teenager, did you never persuade a parent to let you do something they were biased to be predisposed to say no to before you tried to persuade them otherwise?