TO SUM UP:
In all but the harshest treatment and/or conditions, I suspect they'll grow again.
THE DETAILS:
We have Jerusalem Artichokes growing all over the place. They'll grow by accident, in my
experience.
However, they will dry out and won't really grow after they've shriveled-up. So a completely dry environment seems bad. At the same time, we've also kept some in a bucket in
water, and they all rotted. Moist soil ought to work fine, though I'd be wary of muddy conditions. Perhaps being drought-tolerant makes them also susceptible to excessive exposure to moisture.
Keeping them buried under a shallow layer of sand or sandy dirt will preserve them well
enough. Over the winter we buried some in like two or three inches of sand, then laid a surplus window over top of it to make a crude cold frame. They stayed firm for the most part, and I think all of them we hadn't taken in to eat ended up sprouting. They do well after transplanting as well, so long as it's still the early season.
In our experience, pickling and dehydration are two successful preservation methods. But it seems a bit early in the year to dwell on that. Good luck!