>Everyone around here keeps talking about the Cort being renovated, but there has been no mention of it by the Shuberts or anyone else. That's mostly come from posters here thinking it should be renovated, not from any of the powers that be saying it will be renovated.<
It's possible that there will be a limited run tenant at the Cort, but a massive Hard Rock Hotel will be built next door to it, opening in Spring 2019. During some of that time, the Shubert Organization is planning to take the Cort offline for renovations, similar to their recent restorations of the Belasco, Longacre, Booth, Golden, Jacobs and Schoenfeld Theatres.
In the image below, that's the Cort to the left of this behemoth.

Begin at the beginning and go on till you come to the end: then stop.