Illusion Software? The thing HH runs in is Renpy, Open Source.
Or the software he creates the models with? They have any claims on the work created with it?
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They do.
He's using Honey Select to make the pictures and videos, and the 3D assets he's using are not licensed for commercial reuse, including any render thereof.
He got explicit permission for the current monetization model, so that is perfectly fine, but selling the software on, say, steam or GOG... That would run a lawsuit risk.
The situation is even more complicated now, since Illusion no longer exists as a company, since he would have to track down who owns the IP.