Extension of the equilateral conjectures to π/3 tiles

Prove that the divisibility and complete-classification conjectures for equilateral tilings remain valid when the triangular tile has an angle equal to π/3 rather than 2π/3.

Background

The paper develops an analogous construction for integral-sided tiles with one angle equal to π/3. It then explicitly extends both earlier equilateral conjectures—the divisibility assertion and the characterization of possible sufficiently large tile counts—to this setting.