Characterization of the centroid by square preservation

Let $ABCD$ be a square and let $E$ be any point inside it. For each of the triangles $EAB$, $EBC$, $ECD$, and $EDA$, choose the corresponding center $F$, $G$, $H$, and $I$ using the same center function.

Centroid characterization conjecture. If $FGHI$ is a square independently of the choice of $E$, then the common center function must be the centroid.

This asks whether the centroid is uniquely characterized among triangle centers by preserving a square when applied to the four triangles formed by an interior point and the sides of the original square. The supplied text gives no resolution, so the problem remains open.