I was playing as Artanis and I had the Immortal on the board.
My Immortal did NOT gain the stats of adjacent minions at the start of a turn.
My Immortal had at least one mech magnetized to it and had divine shield.
My opponent was Illidan. At the start of the turn, his left-most and right-most minions attacked, and one of them was Lighter Fighter. The Lighter Fighter’s deathrattle triggered and at least one of the hits went on my Immortal (losing divine shield).
The start of turn effect on the Immortal (gain the stats of adjacent minions) never triggered.