Parallelogram characterization for central quadrilaterals of general quadrilaterals

Let $ABCD$ be a general quadrilateral, and let a triangle center be specified by a center function in the angles of each component triangle. Two center functions are equivalent if their ratio is a cyclic function of the triangle angles, so equivalent functions define the same triangle center. Parallelogram characterization conjecture. The central quadrilateral is a parallelogram if and only if the center function is equivalent to

$$\cos B\cos C+k\cos A$$

for some constant $k$. This conjecture seeks a complete characterization of the centers producing parallelograms from arbitrary reference quadrilaterals; the corresponding sufficiency statement is established in the paper, while the necessity remains conjectural.