Yeah, if humidity is right they don't need a log to rub on. Mine shed great. It's always a little rolled up packet, that I have to unroll and check for eye caps because I am anal about checking for eyecaps. They are always found inside of a hide. My set up is such-
I move the water bowl over to the hot side on top of flexwatt and that bumps humidity perfectly during shed.
As far as if you really want to keep the wood, and not have mold, I dont know what to tell you. Because they do require humidity, which will grow mold on wood.
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