= Solution
For the <SLE> driver $d\xi_t=\sqrt6\,dW_t$, apply the <Itô formula> to $\widetilde\xi_t=\phi_t(\xi_t)$. At a fixed spatial point, $\phi_t$ has <finite variation> locally in time by the derivative formula in part (c); there is no extra spatial martingale term. Consequently
$$
d\widetilde\xi_t
=\left(\dot\phi_t(\xi_t)+3\phi_t''(\xi_t)\right)dt
+\sqrt6\,\phi_t'(\xi_t)\,dW_t
=\sqrt6\,p_t\,dW_t.
$$
This exact cancellation is the <target-change locality of SLE6>. For a general parameter the <transformed SLE driving function> would have drift $(\kappa/2-3)\phi_t''(\xi_t)$.
Define the increasing clock and its inverse by
$$
a(t)=\int_0^tp_r^2\,dr=\frac12\operatorname{hcap}(\widetilde K_t),
\qquad t(s)=a^{-1}(s).
$$
The derivative $p_t$ is real and strictly positive before $T$, so this is a valid continuous <time change of a continuous process>. The driver is a continuous <local martingale> with <quadratic variation> $[\widetilde\xi]_t=6a(t)$ and initial value zero. The <Dambis-Dubins-Schwarz theorem> gives a standard <Brownian motion> $\widehat W$ such that
$$
\widetilde\xi_{t(s)}=\sqrt6\,\widehat W_s,
\qquad 0\leq s<a(T-).
$$
If the terminal clock is finite, the Brownian motion may be extended beyond that stopping time; only its stopped part is used here. Reparameterizing the image maps gives
$$
\partial_s\widetilde g_{t(s)}(z)
=\frac2{\widetilde g_{t(s)}(z)-\sqrt6\,\widehat W_s},
\qquad\operatorname{hcap}(\widetilde K_{t(s)})=2s.
$$
These are exactly the defining <Chordal Loewner equation> and <half-plane-capacity parameterization> for $\operatorname{SLE}_6$. The trace is $\widetilde\gamma_{t(s)}$. Hence \b[the image curve is also ordinary chordal $\operatorname{SLE}_6$ after this time change, up to the specified stopping time]:
$$
\boxed{\bigl(\widetilde\gamma_{t(s)}\bigr)_{0\leq s<a(T-)}
\text{ is stopped }\operatorname{SLE}_6\text{ in }(\mathbb H,0,\infty).}
$$
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