You could also run additional 20mm x 20mm beams (called batons?) along the baseboard, just inside where the 9mm side panels will run.
The 20mm x 20mm would be strong enough to glue and screw in place, and you could then glue the 9mm to the baseboard, while also gluing and screwing it laterally to the 20mm x 20mm, giving strength, rigidity, and holding the "floppy" sides in place.
A cheaper and easier way of salvaging your work, I'd say, rather than starting over, which you’re probably thinking about.
Adding lateral bracing beams and painting with some sort of cheap primer will still be required.
Rough diagram: