An Equilateral on a Sliding Segment

In a rectangle $ABCD$ with $AB = 2\sqrt{3}$ and $BC = 6$, points $E, F$ move along $AD, BC$ respectively with $CF = 2AE$. An equilateral triangle $EFG$ is erected on segment $EF$. Find the minimum possible value of $DG$ (over both choices of the side of $EF$ on which $G$ lies and all positions of $E, F$).