In this case I had to do a little more to clean up the paths because of many corners lacking a pixel, though. Specifically I selected color range, then expanded the selection by one pixel, then floodfilled that onto a new layer. Also I had to do a little cleanup of other parts, but that was trivial.

One way to at least somewhat defeat this strategy is to make the paths varying in width. Also, more complex topologies than genus-1 can make things weird as well.