Is there no place in your world for authorities at all? For trust at all? For trusted flawed authorities to become untrusted liar non-authorities in people’s minds?
I have no idea why you are talking about perfect authority.
People can become "untrusted liar non-authorities" in people's minds independent of any actions of their own. The best they can do is their best, and hope not to attract the attention of demagogues.
Of course they can become untrusted regardless of what they do. But the whole question is whether you acknowledge that people also have a major impact through their actions of what people think of them.
What exactly is “doing their best”? Is it lying, trusting the end to justify the means?