The converse ideal-trapezoid tileability conjecture

Let $ABCD$ be an ideal trapezoid, and let $(a,b,c)$ be a triple of pairwise coprime integers satisfying

$$c^2=a^2+b^2+ab.$$

Write $x$ and $y$ for the side parameters of the ideal trapezoid as in the paper. Ideal-trapezoid tileability conjecture. The trapezoid $ABCD$ can be tiled by $(a,b,c)$ if and only if

$$x>c^2-a-b$$

and

$$ab\mid y.$$

The paper presents this as the converse to an earlier sufficient tileability proposition. Proving it is identified as an intermediate step toward the equilateral divisibility conjecture; necessity is not established in the supplied text.