Divisibility of equi-constructible side lengths

Prove that every side length X of an equilateral triangle tileable by congruent copies of an integral-sided triangle (a,b,c) with c^2=a^2+b^2+ab and an angle 2π/3 is divisible by ab.

Background

For square-free a and b, the paper proves that an equi-constructible side length X must be a multiple of ab. It conjectures that the same divisibility holds without the square-free restriction, and gives (5,16,19) as the smallest interesting test case.