A question about an item in chapter 9

I finished the game first without obtaining the bracer then I started chapter 9 again from the chapter select menu and got the bracer.

But after that I realized that whenever I start any chapter from the chapter select menu say chapter 10 or 11, the bracer is gone.

Does that mean that it will also not carry over to ng+ since I got it through chapter select and not during my original playthrough?