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Cork only. Read how it works and you'll see that it can't work with a screw top. You can use Private Preserve with screw tops.

Do you know how private preserve compares to just pumping the air out, which is what I currently do?

My feeling is that PP works better. I think this because the gas displaces all the oxygen, where pumping the air out only removes some of it.

Best idea is to try a can for yourself.