We've not overwatered it (we don't think) and have owned these plants for two years. In the space of a week these started to wilt.
The right hand plant has a soft stem at the bottom.
Is it possible to transplant these?
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These are Yucca Gigantea. They are really tough plants that will survive a lot of neglect and even low light. When grown in full sun with a light hand on the watering the foliage is a dark green and the stems can be as thick as your arm
These plants must have good drainage or the stems get soft and rot. Excellent light from a south facing window is a good idea as well.
It would be a good idea to repot them into a soil less mix for cacti or succulents as that drains better. You could probably put both into one smaller pot than they have now. A smaller pot dries out faster.