Icons need to be easily recognizable. A big part of that is the design of their attire. Could you picture the Caretaker in a white surgeon's outfit? How about Jack in street clothes without makeup on? If the icon doesn't look like the icon, it loses its purpose. That's why Lady Luck won't go anywhere without that green dress of hers. I'm hesitant to say we'd see her without the big wheel, either.
That being said, can you see that green dress in the trenches? It's a real scene-killer.
It's the same with all the house environments this year. She wouldn't fit in any of them. That's why I'm betting you won't see anyone playing her outside of her scarezone.