Aperiodic Monotile Puzzle

The aperiodic monotile discovery prompted some reverse engineering, snapping a path to regular hexagons for the proper lengths and angles without mathing too hard:

The resulting red path is the “hat” monotile, here shown as a PNG for neatness:

With SVG hat in hand, I laid and cut a trial puzzle based on the sample shown in the paper:

Which looked promising enough to add a few rings around that layout and turn it into an actual, albeit low budget, puzzle:

The paper notes that one can build mutually incompatible patches, which is the state I immediately blundered into:

The upper and lower left halves cannot be combined to extend rightward, as the middle section is incompatible with both. I began growing patches from the upper and lower right corners, hoping to use them to rectify the left half, but producing a small un-fillable situation in the middle:

Obviously, I need a cheat code. I’m resolutely not looking at the source layout for a while.

This would surely look good in fluorescent edge-lit acrylic!

The LightBurn SVG layout as a GitHub Gist: