The escaping-trajectory conjecture for isosceles triangle tilings

Let an isosceles triangle tiling be the tiling obtained from congruent isosceles triangles, and let its vertex angle be $\theta$.

Isosceles escaping-trajectory conjecture. An isosceles triangle tiling has an escaping trajectory if and only if its vertex angle is not of the form

$$\theta=\frac{\pi}{2n+1}$$

for some positive integer $n$.

The preceding theorem establishes that every trajectory in an isosceles triangle tiling is either periodic or drift-periodic, while the equilateral case is known not to have a drift-periodic trajectory. The conjecture asserts that these are the only exceptions.