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Your plant is not J. podagrica but another member of the genus, and I think there is a good chance it will be recover. The soft part will need to be cut away. Use a sharp knife and cut through healthy tissue outside the area of rot. You want healthy margins everywhere when you're done. Usually the color is a real giveaway. Wipe off and dip your knife in ethanol in between cuts if there is any chance of contaminating healthy tissue with rot. And give the plant a week or so out of soil to recover. When you make the cut, try to do it in a way that does not trap water, or otherwise make the caudex vulnerable to rot in the future. Avoid big empty hollows. After it is fully healed (maybe a week out of soil and then a week in soil) you can water lightly.
