The laminated particle-board cabinets that will not hold mounting screws can be successful when used with "binding posts" or "Boston-posts" type of fasteners. It's the only way I could mount addt'l bath-towel racks on the bathroom door.
Available online or at hardware stores in various lengths.
I bought the same Simple Human paper towel holder and my husband mounted it vertically on the side of the cabinet next to the door....he also made a aluminum base which fits just under the towel. It works perfectly except when we're driving and it unrolls, but I solved that by using a small paper clamp holding the rolls closed. Love the location.
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