During one of my flights, I had a larger than usual carry on, which made for a tight fit, especially with my jacket. They suggested and offered to store it in the closet in front, where I got the impression there was ample space and may store things considerably larger than the seat closet. This was in 2A/K, where I think it's larger than 1A/K?

Assuming your box fits, I'm sure they can store it there.

PS - I'd assume it's safer in the FA closet, unless you were planning to stay awake the entire flight. There have been a few times when traveling with some very important documents that I stayed awake the entire flight just to keep an eye on it.