A generalized conjecture about cutting a polygon into triangles of equal areas

In 1970 it was proved that a square cannot be cut into an odd number of tri angles of equal areas. In 1990 it was proved that any centrally symmetric p olygon has that property. In the present paper we propose a broad generaliz ation, which would also imply that any polygon whose edges are parallel to the x- or y-axes has that property. We prove that the generalization holds for polygons with at most six sides.