The tile-count divisibility conjecture for $(2\alpha,2\beta,\alpha+\beta)$ targets

Let $(a,b,c)$ be the side lengths of the triangular tile, with angles determined by $a,b,c$, and let $T$ be a triangle with angles $(2\alpha,2\beta,\alpha+\beta)$. Tile-count divisibility conjecture. If $T$ can be tiled by copies of $(a,b,c)$, then the number of tiles has the form

$$(a+2b)(b+2a)m^2$$

for some integer $m$. Any counterexample must satisfy $a\equiv b\pmod 3$.

The preceding lemma proves this form under the hypothesis $a\not\equiv b\pmod 3$. The construction in the paper supplies infinitely many tile counts of the displayed form, while the exceptional congruence class remains the possible source of counterexamples.