 Solution

For every bounded simply connected $D$, translation $z\mapsto z+x$ is a conformal map from $D$ to $D+x$. Conformal restriction says that translating ${\rho }_D$ gives ${\rho }_{D+x}$. Exhausting the plane by such domains proves that the whole measure $\rho$ is translation invariant. Consequentlywhere $T_x(\gamma )=\gamma +x$ and ${\mu }_x$ restricts $\rho$ to loops surrounding $x$.