I believe that must be what TRUMP wants him to do, to put it to rest.
Why he cares, I do not know. Not in my wildest dream would TRUMP pick Cruz for VP. Jeff Sessions, maybe, but not Cruz with the immigration baggage and the TPA kabuki, and the increased numbers of foreign high skilled workers, and Cruz yakking it up about “pathways to ‘legalization’ “ as a reward to invaders. TRUMP can not go to Cruz on his (TRUMP) signature issue.
Trump doesn't trust him. He won't/wouldn't pick him. That goes both ways, for different reasons. The agendas are akin to a clash of civilizations, literally.
The birther issue dies as Cruz falls in the polls. It's only important to the extent he has a chance at the nomination.