= Solution
Set $g_K(iy)=u_y+iv_y$. The <hydrodynamic normalization at infinity> gives
$$
u_y=O(y^{-2}),\qquad v_y=y-\frac{a_K}{y}+O(y^{-2}),
\qquad \frac{v_y}{y}\to1.
$$
The real interval under consideration lies outside the hull, so the reflected boundary map sends it monotonically to $(g_K(x),g_K(b))$. <Conformal invariance of planar Brownian motion> transports its <harmonic measure> to the half-plane. By the <Poisson kernel for the upper half-plane>,
$$
\mathbb P_{iy}(B_T\in(x,b))
=\frac1\pi\int_{g_K(x)}^{g_K(b)}
\frac{v_y}{(r-u_y)^2+v_y^2}\,dr.
$$
The endpoints are fixed and finite. On this interval,
$$
\frac{y v_y}{(r-u_y)^2+v_y^2}\longrightarrow1
$$
uniformly, using $v_y/y\to1$ and $u_y\to0$. Integrating gives \b[the interval-length limit]:
$$
\boxed{\lim_{y\to\infty}\pi y\,\mathbb P_{iy}(B_T\in(x,b))
=g_K(b)-g_K(x).}
$$
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