Stein's equidissection conjecture for balanced polygons

Let $B$ be a plane polygon with clockwise oriented boundary. The polygon $B$ is balanced if its edges can be divided into pairs such that, in each pair, the edges are parallel, equal in length, and have opposite orientation. A polygon is cut into triangles when it is presented as a union of finitely many triangles whose interiors have empty intersection.

Stein's conjecture. A balanced polygon cannot be cut into an odd number of triangles of equal areas.

This conjecture generalizes results of Monsky on equidissections of squares and centrally symmetric polygons. The paper proves the corresponding result for lattice balanced polygons of odd area, while the unrestricted conjecture is not resolved here.