Macro-tile structure of non-commensurate tilings

Prove that, in the six specified non-commensurable-angle configurations involving a tile angle of 2π/3, the tiled triangle T can be partitioned into ideal trapezoids and triangles similar to the tile R.

Background

The paper’s constructions use ideal trapezoids and triangles similar to the tile as recurring intermediate pieces. The authors elevate this observed structure to a conjecture covering the relevant non-commensurable cases.