Yes they will.
They usually all fight to lay in the same ONE every day.
It is very rare that I find eggs in more than one nesting box of mine.
They all lay in the same one.
You do not need to think about this now...my guess is you will start getting eggs late December or early January...but come next summer...those nesting boxes are going to be too hot.
Awesome work!
The only problem I see is the run is not fully predator proof...so I worry that cutting the one coop wall out and there not being a way to secure them up at night might be a problem.
If you are planning on digging down and burying hardware cloth (an apron) you fold it out/away from the wall, think L shape so you don't necessarily need that bottom piece of wood.
You could use zip ties to attach the L shaped hardware cloth apron to the fencing.