Being that the coop is not that big. And depending on how many chickens you have, and whatever the temp might be on any given day/night. I believe that yes, you may have some that will prefer to roost out in the protected run. The run looks pretty secure, so that shouldn't be a big deal.
Jack
If you are totally confident in your run's construction, you could just leave their door open, or just doorless. Then you don't have to make a trip every morning to let them out. And really, I don't think you have to worry much about having the pop door open in the winter. Should not be a big...