The $4n+2$ period conjecture for triangle tilings

Let a triangle tiling be a tiling of the plane by congruent triangles. The period of a periodic trajectory is the number of its trajectory segments in one cycle.

$4n+2$ period conjecture. In a triangle tiling, every periodic trajectory has a period of the form

$$4n+2$$

for some integer $n$.

This conjecture concerns the possible periods of periodic trajectories in triangle tilings; the supplied text gives no proof or resolution.