Naturally especially because of the clues pointing to a faked death, IMO. Besides that, Rand and LTT are different personalities. If LTT dies, Rand won't die at all.

However, I think of course that the conflict between Rand and LTT will come to a climax in the end. Perhaps the DO will even tempt LTT in the Pit of Doom to switch sides by promising him to give Ilyena him back or something like this, so that LTT tries to control Rand's body. But how Rand will actually solve the LTT problem is one of those things I'm not yet sure. Perhaps LTT will simply vanish when he isn't needed anymore after TG, like e.g. ta'veren as an effect of the Wheel/Pattern can appear and vanish.