= Solution
Write $h=h_U+h^{\mathbb D\setminus U}$ with $U=B(0,e^{-t})$. By part (b), the harmonic part contributes the same value $h^{\mathbb D\setminus U}(0)$ to every inner circle average. At radius $e^{-t}$ the zero-boundary part contributes zero; equivalently, take the limit from inner circles and use the assumed continuity. Hence
$$
X_s-X_t=(h_U,\rho_s).
$$
The field $h_U$ is independent of $h^{\mathbb D\setminus U}$, so the increment has the required independence.
The dilation $z\mapsto e^tz$ maps $U$ onto the unit disc and maps the circle of radius $e^{-s}$ onto the circle of radius $e^{-(s-t)}$. The <Conformal invariance of the two-dimensional Gaussian free field> therefore gives
$$
X_s-X_t\overset d=X_{s-t}.
$$
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