= Solution
The <Conformal Markov property of SLE> says that, conditionally on the curve through time $t$, mapping out the initial hull by $g_t-U_t$ turns the future into an independent <Schramm–Loewner evolution> in $(\mathbb H,0,\infty)$. More precisely,
$$
\widetilde K_s
=\bigl(g_t(K_{t+s}\setminus K_t)-U_t\bigr)^{\mathrm{fill}}
$$
has driving function $\widetilde U_s=U_{t+s}-U_t$. Since $U_t=\sqrt\kappa B_t$, the <stationary increments> and <independent increments> of Brownian motion give the asserted independence and equality in law.
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