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Help with Self-Intersecting Sweep around complex path

Hi there,
I'm looking to sweep a sketch along the curves of this polyhedron, so that every curve turns into a matchstick. While Sweep lets you do all six sides of a hexagon here, it fails as soon as you click a branching sweep path from there. The error says that it is self-intersecting. I don't mind that it intersects, because it's putting material right where I want it, and it might as well just boolean all swept paths into one part. I know that mathematically it can't handle two of the same thing in the same place though.

One solution I tried is to generate a polyhedron a little larger than the first one, and making and thickening surfaces that bridge between the inner and outer polyhedron, but this is too time-consuming as you have to bridge every matchstick manually.

Has anyone got any ideas about how to turn every line of this polyhedron into a thin stick of material, to make a single part?

https://cad.onshape.com/documents/113dc998982b38225131308d/w/7f0af6aacf07771e89bad19b/e/69a643cf667145898976489a

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