Orthodiagonal characterization for central quadrilaterals of general quadrilaterals

Let $ABCD$ be a general quadrilateral, and let a triangle center be specified by a center function. The central quadrilateral is formed by placing the chosen center in the four component triangles determined by the diagonals. Orthodiagonal characterization conjecture. The central quadrilateral is orthodiagonal if and only if the center function is $1$, that is, the center is $X_1$. This would characterize the unique triangle center producing an orthodiagonal central quadrilateral for every general reference quadrilateral; the paper proves the sufficiency for $X_1$, but the necessity is left open.