I don't have a quantitative argument, but my intuition is that it might still be possible to make a track that globally has no net tension, yet still has "local

I don't have a quantitative argument, but my intuition is that it might still be possible to make a track that globally has no net tension, yet still has "local" tension somewhere. This might be done by creating a shape that slightly intersects itself, pushing that section into tension, while a complementary section sums to the negative of the first section but without self-intersection.