BALTIC-WAY 2023 Problem 14

Let $A B C$ be a triangle with centroid $G$. Let $D, E$ and $F$ be the circumcentres of $B C G, C A G$ and $A B G$, respectively. Let $X$ be the intersection of the perpendiculars from $E$ to $A B$ and from $F$ to $A C$. Prove that $D X$ bisects the segment $E F$.