I am going to be getting a couple pigeons this spring and I will be keeping them with my chickens (I'm taking all of the special precautions just so you know). My question is: Will 2"x4" wire mesh fencing (whatever it's called) keep them in? Will they squeeze out? Will they even try to squeeze out? My run is very big, about 22'x16' and 6' high with branches and perches to sit on. And yes it has a wire roof.