 Solution

Put $A=\mathbb{H}\setminus U$. The neighborhood assumptions at zero and infinity mean that $A$ is a bounded hull away from the starting boundary point. For $t<T$, the conjugacy map $h_t$ is the hydrodynamically normalized mapping-out function of the remaining forbidden hull after removing $K_t$. Its closure avoids the driver, so the boundary derivative is an excursion avoidance probability yields $0<{\Sigma }_t\le 1$. Thus $0<M_t\le 1$, and the bounded local martingale criterion makes the localized martingales uniformly bounded and uniformly integrable.

Choose localization times ${\tau }_n\uparrow T$, also bounded by $n$. Optional stopping gives $\mathbb{E}{M}_{{\tau }_n}=M_0$. The supplied terminal limit and continuity of the power function giveSince these variables are bounded by one, dominated convergence yieldsAs usual for a chord starting at zero, the time-zero point is understood as the boundary prime end of $U$; the avoidance assertion concerns the subsequent curve. The boundedness step is what upgrades the local martingale calculation to an equality of probabilities.