The divisibility conjecture for equiconstructible equilateral triangles

Let $(a,b,c)$ be the side lengths of the triangular tile, and say that a positive real number $X$ is equiconstructible by $(a,b,c)$ when an equilateral triangle with side length $X$ can be tiled by copies of that triangle. Divisibility conjecture. All equiconstructible $X$ are divisible by $ab$; in particular, for the tile $(a,b,c)=(5,16,19)$, every equiconstructible $X$ is divisible by $16$.

The preceding lemma proves the divisibility statement when $a$ and $b$ are square-free, while the case $(5,16,19)$ is presented as the smallest interesting case beyond that result. The conjecture is intended to constrain the possible side lengths of equilateral tilings.