I guess it depends on how wilted they are but I've found that they are pretty tough to kill. The best storage is in the ground attached to the plant. Every time I've tried to store them, they last a week at most and then they go soft and are best suited for pigs or replanting. It's best just to harvest what you plan on eating within the next day or so. Once the ground freezes, you're kinda locked out unless you mulch super heavy around the plants. I'd plant whatever you're not comfortable eating and don't be surprised if you have a forest of Sunchokes again next year.
I planted six tubers 4 years ago and now I have a hedge of them. The first year yielded about 10 gallons of tubers. I fed most of them to the animals and then replanted about a gallon of them after they went brown and wilty. Since then I've been able to pull about 5 gallons of tubers per 2 feet of hedge without decreasing the next years yield. As a matter of fact the "occupied space" expands annually. The hedge is about 3 feet wide and 30 feet long or so and growing. Animals eat the whole plant too so that's a plus. I'm not a fan of them actually but they make a nice wind break and nutrient trap.
I have heard that you can extend their life span by storing them in wet sand, or in slightly damp soil. Which makes sense, they naturally live in the soil all winter, so if you can replicate that environment, they will be eating quality much longer. I will be experimenting with that this winter. I have some in a soil/sand mixture. I have been told that they should last a few months with this method.
I bought a kg on the internet, they came wrapped in newspaper. I put them behind the door in the washhouse, out of the sun, where it would have been cool. I planted them several months later and they were not soft. I also had some yacon stored in the same place, which were in a plastic bag within the newspaper. The yacon were rotten, because of the plastic. Planted some of them out, and they still took.
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