Does immunity protect one against lying or perjury?
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It does if none of your co-conspirators tells the truth and the agency refuses to give up documents. Besides there’s always a presidential pardon and a good paying “non-profit” job just waiting to happen in the back of their minds.
I have proposed that there should be an amendment to the constitution which prohibits the president from pardoning anyone who is not an “arm’s length” from the president or his administration. If it is shown that the pardon was used to protect the president from proscecution, the pardon is revoked, and the sentence doubled.