I think the Exarch is going to die but not stay dead.
Given how the last couple patches have been about research into the soul and developing of special Auracite so that the Scions could be safely returned to their vessels, to turn around and say "just kidding, the Exarch dies and that's it, that's how we solve it, moving on now~" would be a massive slap in the face for the past year and a half of distractions if that research goes unused. Particularly since everything after the climax of Shadowbringers has been "How do we save the Exarch?"
But we do know that the soul infusion process they were going to use would be the same as what Unei and Doga used on G'raha to transfer control over the Tower, and that the Exarch was experimenting with that same process himself with the auracite. And that whatever he did to extend his life with the Tower, he's only got a matter of time even if Elidibus doesn't kill him.
My theory is that the Crystal Exarch is going to die on the First, which will allow the Scions to be returned, but -- given how much importance has been placed on the ties between memory and soul this expansion (including us surfing memories in the Bozjan storyline, the memory of Amaurot, etc), and the emphasis on "memory" in the trailer -- that he will have infused enough of his soul into the auracite that it can be used as a vessel for his own memories to be transferred into another vessel, and that he will give us a means to re-awaken the Crystal Tower to get inside.
Then we'll go back to the Source and give the auracite to G'raha so that we can have a pseudo-Exarch who lives on through him without having to make the same sacrifices. Then G'raha will be honorarily inducted into the Scions and have his own arc about coming to terms with having two lives, maybe even having a bit of a bond with his other soul like we did with Ardbert or Ryne did with Minfilia.
(Something else worth considering is that reawakening the Tower could then lead to an expansion about Meracydia, including what we learned of it during the Warring Triad story. Food for thought.)