Nah, he has the same immunity other executive offices have had in past rulings. This is the point, there is nothing about this immunity different than what I spoke about previously. Not only that, pretending that it covers everything like ordering illegal acts from a General is idiocy and covered in the ruling where ruling on what is "official" or not is left to the lower courts.
Was calling the Georgian Governor official? We don't know yet, there has been no ruling on that. Though you don't get to assign a motive to it, and if it is decided to be official what was said in the call cannot be used against him as evidence... it has not yet been decided to be "official".