Here's a simple box that can be mounted if you like. I didn't think to put a slot in the bottom for easier single towel removal.
I printed this with PLA but PETG will work also. I actually prefer PETG over PLA but it prints slower. I also chose 0.3mm draft to speed this up.
I'd use a flex material for the caps since they'll fit snugly. If you print them with PLA or PETG I'd scale them up just a hair. You can use a glue too if you like. I found a glue stick works great because you can still remove them later if you need to relocate the box.
The author hasn't provided the model origin yet.