Is it P. dolabriformis? Mine did that a while back and suddenly the trunk was woody. But yes, I think it would be a lot more thirsty than the Lithops, but never had one by itself to have any basis for comparison. For your container that has those 2 together, I might use a squirt bottle to spritz a bit of moisture at the base of the Pep, but not enough to get over to the Lithops side. The mini garden my Lithops is in has a lot of other plants in it, but I pour a lot more water near/over/around all but the Lithops, which is kind of by itself right in the front. Someday it will need to be redone, and I'll see where the roots have gone from each thing, and how big they are...
I've thought about hanging that wire thingie next summer. IDK if our ceilings are up to it inside though! It's seriously heavy after I watered it. There may or may not be some Peperomia in there. My eyes aren't good enough to know from the pic, I'd have to go look, I just put stuff in there until I got sick of doing it (TWICE, once in the top, then later on 2 sides.) LOL!