I had four variations on trimmer line here of the larger (2.2mm+) diameter to test for this.

• The regular round stuff.  I can't imaging that would work too well.  It will slide in both directions.
• Star shaped, with raised edges and recesses along the length.  Better.  The issue here is that the star shape was never extruded perfectly.  Both the stuff I had, and the same stuff I found brand new in the store, had some ridges along the length.  It will not slide well in the long dimension as such.
• Octagonal shaped line I used for the proof of concept.  It was pretty good.. smooth along the length for the most part, but still had some defined edges which would grip the ground when driven off the long axis.
• Square line.  This seemed like a good idea, until I tried to bend it.  The square shape meant it was impossible to keep a corner facing down.  It always wanted to align so one of the flat surfaces was the one which would be hitting the ground.

So I went looking, and I found this.

It has an overall triangular shape, which means that when you bend it into a circle, one point of the triangle is always facing the outside of the bend.  That was something I didn't really expect, but it kinda makes sense in hindsight.  The line resists the bending more where it is thicker, and so aligns with a thin edge out. It's also perfectly smooth along the length.

They even have a massive 4mm diameter version of this same line in stores locally, so my upper limit for the filament diameter just went away.  About 10 of those, spreading the weight over 4 wheels, could probably support a human being with the spring tension of the line itself.

I'll be using this going forward.