I searched and I didn't find any answer (positive or negative):

Suppose I have a group $G$, and a group family of probability measures . (i.e, there is a probability measure $P$ on $G$, and we define $P_g(A)=P(gA)$ as $g$ in $G$ is a parameter). Also this family is assumed to be complete.

The maximal invariant statistic is always ancillary. Is it always maximal ancillary? Is it always the greatest ancillary?

Are there any reference to a proof or a counter example?