= Solution
The <conformal automorphism of the upper half-plane> with the indicated three <boundary> values is
$$
\boxed{\Phi(z)=\frac z{z+1}.}
$$
Its <Möbius transformation> matrix has determinant one, so it preserves $\mathbb H$. The inverse is $\Phi^{-1}(w)=w/(1-w)$. In particular
$$
\Phi\bigl(( -\infty,-1]\bigr)=(1,\infty]
$$
on the extended real <boundary>. Therefore the image trace's first contact with the latter ray corresponds to the original time $S$. The map sends the original target at infinity to the new target $1$; it does not preserve the target of the chordal curve.
We use the image <half-plane capacity> as its time parameter:
$$
u(t)=\frac12\operatorname{hcap}(K_t^*)=\int_0^t\Phi_s'(\xi_s)^2\,ds,\qquad t<S.
$$
Since $\Phi_t$ is a <conformal automorphism of the upper half-plane> analytic at $\xi_t$, its real <derivative> there is strictly positive. Hence $u$ is continuous and strictly increasing before $S$. The original time $t$ and image time $u$ must be kept distinct throughout the comparison.
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