Armor priorities for an anti-aircraft battleship

Would an anti-aircraft battleship retain full battleship armor?

It might retain strong deck protection without needing the full traditional belt, although converting an already-built Iowa hull would leave much of its armor in place.

Protection against air attack differs from the belt armor intended for battleship gunfire. A purpose-built anti-aircraft ship might emphasize deck protection and accept less belt armor, while a conversion of an existing Iowa hull would naturally retain structure already built into it. The exchange shows how mission and starting hull constrain an apparently simple refit choice. Protection should follow the expected threat rather than reproduce a traditional battleship scheme by default.