As I understand it, the United States limits are 22x9x14 inches (559×229×356 mm).  I have a box for my computer monitor box that is 22x3x15 in (559×76×381 mm), so it does not comply and I would expect it to be forced into check-in.  I would prefer to carry it by hand.  

How strict are the US authorities with the 14 inch (356 mm) limit?

That being said, would it be the TSA that would that would force the issue/prevent me from boarding with a 22x3x15 in (559×76×381 mm) monitor box?   

I suspect the airline would be indifferent.