I tried one more time. Over-extruded at the start as I didn't calibrate, but I was just interested in testing the pop-through mainly, so I just let it finish.
So... the one on the left has a steeper angle on my accordion fold. Basically as far as I can push the overhang in TPU without supports, about a 45 degree angle.
It's also mixed material, with the first .5mm being 50/50 ABS/TPU, so not something your average user could produce anyway.
This stiffened it up enough to maintain it's shape and allow the accordion to work, but the steep overhang angle makes it look terrible. I refuse to give in to supports!
I wish I could get this sorted. If this worked, I could print the entire top as a single, water-resistant unit (or two, but just limited by printable area).
Edit: Looking closely at my success and failure here, I noticed that the corners work well while the straight sections.. don't. So I'm starting over in OpenSCAD and making one that is all corners... aka... round. :)