Golden-parallelogram maximal-denominator conjecture for three-move pieces

Let a piece have exactly three moves, and consider the vertex configurations arising from its golden parallelogram constructions. Three-move golden-parallelogram conjecture. For a piece with exactly three moves, one of the golden parallelogram configurations gives a vertex with the largest denominator. The paper uses this conjecture to motivate formulas for the largest denominators of particular three-move pieces, but gives no proof of the general maximality assertion.