Hi TIA, I'm not a cabinet maker but I am handy. I took a look at our custom cabinets. They use side mounted drawers, not under-mounts; but they are essentially the same concept. Each drawer stack is one box without any dividers. I had to look up your hardware and almost immediately found a nice installation video. It also shows a box without dividers (eventually). Each of our custom cabinet units was made from a complete box butted to the next box. Therefore most 'sides' are actually almost 1.5 inches thick and therefore massively stable.

Here is a link that might be useful: Blum install