Uniqueness of the centroid-preserving triangle center for arbitrary quadrilaterals

Let $ABCD$ be an arbitrary quadrilateral, and let $X$ be a triangle center. Let $E$, $F$, $G$, and $H$ be the $X$-points of $\triangle BCD$, $\triangle CDA$, $\triangle DAB$, and $\triangle ABC$, respectively.

Centroid-preservation uniqueness conjecture. The centroid of $ABCD$ coincides with the centroid of $EFGH$ if and only if $X=X_2$.

This is posed as an example in a request to determine whether a relationship found by searching the first 1000 centers is unique or whether further centers satisfy it. The source provides no proof or resolution.