= Solution
Work with a locally growing <Loewner chain> started at $0$, with $A_0=\varnothing$ and the <half-plane-capacity parameterization> $\operatorname{hcap}(A_t)=2t$. The <Conformal Markov property of SLE> combines invariance under conformal changes of the marked domain with restarting after an initial hull has been removed.
In the half-plane this means, first, that $(r^{-1}A_{r^2t})_{t\geq0}$ has the same law as $(A_t)_{t\geq0}$ for every $r>0$. Second, conditionally on $\mathcal F_t$, the future hulls mapped by $g_t-U_t$,
$$
\widehat A_s=(g_t-U_t)(A_{t+s}\setminus A_t),\qquad s\geq0,
$$
with the usual hull closure, have the original law and are independent of $\mathcal F_t$. Their capacity is $2s$ by the <half-plane-capacity composition rule>. One may require the same restarting property at finite <stopping times>.
Equivalently, the driver has the scaling law and conditional increment law
$$
\boxed{(r^{-1}U_{r^2s})_{s\geq0}\overset{d}=(U_s)_{s\geq0},\qquad
(U_{t+s}-U_t)_{s\geq0}\mid\mathcal F_t\ \overset{d}=(U_s)_{s\geq0}\text{ independently}.}
$$
The conformal invariance clause matters: the restarting property alone would also allow a Brownian driver with deterministic drift.
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