Yes, they will reload if there’s still any nectar flow on…so you have to remove the super from the hive. Problem is you can set off robbing from other nearby colonies. Others will hopefully weigh in, but the only way I’ve done it is to set the super on a table well away from any hives. Cleanup activity can be pretty frenzied for a day or so, but there hasn’t been any spillover to my hives (they’re raiding together on neutral shores)
Some folks then give the frames a rinse, and then enclose them in plastic, then keep them stored in the Flow super stashed somewhere cool, dry, & protected from critters.
Personally, mine stays on a wooden tray that seals off the bottom, with a sheet of parchment paper under an inner cover and topped by the peaked Flow roof, out on my side porch all winter.
I replace the peaked roof with an insulated cover on the hive it was on.