= Solution
With the notation supplied in the question, $\widetilde U_t=\psi_t(U_t)$ and $dU_t=\sqrt\kappa\,dB_t$. The <Itô formula> and $\partial_t\psi_t(U_t)=-3\psi_t''(U_t)$ give
$$
d\widetilde U_t
=\sqrt\kappa\,\psi_t'(U_t)dB_t
+\left(\frac\kappa2-3\right)\psi_t''(U_t)dt.
$$
For $\kappa=6$, the <drift> vanishes. The resulting <continuous local martingale> has <quadratic variation>
$$
d\langle\widetilde U\rangle_t
=6\psi_t'(U_t)^2dt
=3\,d\widetilde a(t).
$$
If $s=\widetilde a(t)/2$ is the usual half-plane-capacity time, then $\langle\widetilde U\rangle=6s$. The <Dambis-Dubins-Schwarz theorem> therefore gives
$$
\widetilde U_s=\sqrt6\,\widetilde B_s
$$
for a standard <Brownian motion> $\widetilde B$. The mapped hulls are consequently an $\operatorname{SLE}_6$, which proves locality.
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