Solution
= 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. Consequently
$$
\mu_x=(T_x)_*\mu,
$$
where $T_x(\gamma)=\gamma+x$ and $\mu_x$ restricts $\rho$ to loops surrounding $x$.