Maybe you could paint the wall greenscreen, and hide it behind a tapestry when not in use. Then you wouldn't need a place to store or deploy the screen frame hardware.
edit: froztbyte has a good point! To clarify, I mean only an appropriate square of wall that is in view of the camera. Also, I'm a deranged person that consumes pivot via podcast, so I don't know the layout of the room. A more clever person than I would have checked on that before offering a potentially impractical suggestion.
I'm enjoying the mood today. We're all looking for what the next Big Dumb Thing will be that we'll be dunking on next year, like we're browsing the dessert menu at a fancy restaurant.