Cyclic characterization for centers of tangential quadrilaterals

Let $ABCD$ be a tangential quadrilateral, and form the central quadrilateral by placing the chosen triangle center in the four component triangles determined by the diagonals. Cyclic characterization conjecture. The central quadrilateral is cyclic if and only if the chosen center is $X_1$. The paper cites a proof of the forward direction for $X_1$, but the asserted uniqueness of the center remains open.