= Solution
Let $t(u)$ be the inverse image-capacity clock. Part (a) and the <quadratic variation of a stochastic integral> give
$$
\widetilde\xi_u=\xi^*_{t(u)},\qquad [\widetilde\xi]_u=6u.
$$
The <Dambis-Dubins-Schwarz theorem>, or the <Lévy characterization of Brownian motion>, now shows that $\widetilde\xi_u/\sqrt6$ is standard <Brownian motion> up to the terminal image time. If that terminal clock is finite, it can be extended by a fresh Brownian continuation; this does not assert anything about the subsequent image trace. The initial value is zero because $\Phi(0)=0$.
Differentiating the image <Loewner chain> with respect to its own <half-plane capacity> time gives
$$
\partial_u\widetilde g_u(z)=\frac2{\widetilde g_u(z)-\widetilde\xi_u},\qquad
\widetilde g_u=g^*_{t(u)}.
$$
Thus, until its first hit of $[1,\infty)$, the reparametrized image $\Phi(\gamma)$ has the ordinary chordal <SLE> law from $0$ to infinity. This follows from the uniqueness of the <Chordal Loewner equation> for a given continuous driver; the continuous <Loewner trace> is recovered from the inverse-map limit. Localization can be removed by exhausting times strictly before this <boundary> contact. The terminal event is defined from the trace itself, so equality of the stopped trace laws follows as well.
On the other hand, the definition of <Conformal invariance of SLE> says that the full unparameterized image $\Phi(\gamma)$ has the law of chordal $\operatorname{SLE}_6$ from $\Phi(0)=0$ to $\Phi(\infty)=1$, namely the law of $\gamma'$. Its stopping event at $[1,\infty)$ is exactly the image of the original stopping event at $(-\infty,-1]$. Therefore the same stopped image has both laws: ordinary $\operatorname{SLE}_6$ from $0$ to infinity stopped at $T$, and $\operatorname{SLE}_6$ from $0$ to $1$ stopped at $T'$. We conclude
$$
\boxed{(\gamma_t:0\leq t<T)\ \overset d=\ (\gamma'_t:0\leq t<T')
\quad\text{as curves up to increasing reparametrization}.}
$$
The statement does not identify $T$ with $T'$ in their original clocks. It is <target-change locality of SLE6>, derived from the zero drift and capacity time change rather than assumed as a theorem.
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