I'm not even sure of what he wants immunity for. The actual taping of the Rams violated no criminal laws, even if it happened. Are the Rams going to sue him or his then-employer, if they sue anyone? No one sues an employee when there are deeper pockets. If he was doing it at the instruction of his employer, even his employer can't sue him.
The only thing I can figure is that he wants immunity from criminal prosecution for stealing the tape(s) from the Patriots' video room. He wants immunity from a misdemeanor charge of larceny under $250?

They couldn't even extradite him from Hawaii to face those charges.
