Of course you should assign seats! I know you love us, so let us all stop acting like cattle! But assign seats differently from all the others. Loyal SWA flyers should just check in as now, internert or in person, but instead of getting a simple A ,B or C boarding passs, we click to a second screen and pick a specific seat number. Same priority but less confusion. Then you keep, or even improve, boarding efficiency by boarding by seat number sections from the rear of the aircraft first. The boarding pass could list both a seat number and a boarding priority (I recommend at least 4 sections - A, B, C, D) and the rear-most section would load first and so on to the front. The cabin attendants would need (for a while) to remind continuing PAX not to change seats, and they would need to watch that early boarders don't fill the front luggage compartments, but those things should be "relatively" easy to police. Good luck with your test, I hope it works.
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