The commensurable-sides conjecture for $2\pi/3$ tilings

Let a non-reptile tiling have incommensurable angles and let the tile angle denoted by $\gamma$ satisfy

$$\gamma=\frac{2\pi}{3}.$$

Commensurable-sides conjecture. The tiling must have commensurable sides.

The paper notes that most constructions use an equilateral triangle as an intermediate object, for which commensurability is known, and suggests that a construction violating the conjecture would need to avoid such an intermediary. The conjecture remains open in the supplied text.