Uniqueness of the orthocenter for area-preserving central quadrilaterals

Let $ABCD$ be a general quadrilateral, and form the central quadrilateral from the four half triangles determined by its diagonals, using the same triangle center in each. Area-preserving center uniqueness conjecture. The center $X_4$ is the only center such that the central quadrilateral has the same area as the reference quadrilateral for all quadrilaterals $ABCD$. The paper proves area equality for $X_4$, but the assertion that no other center has this universal area-preserving property remains open.