Finite conformal exit clock for a power wedge ($A_T={\int }_0^T{\alpha }^2|B_s{|}^{2\alpha -2}ds<\infty (0<\alpha \le 2)$)

For Brownian motion started at $i$ in the upper half-plane, the real-axis hit $T$ is finite and its Cauchy distribution is nonzero almost surely. The continuous path on $[0,T]$ therefore stays in a compact annulus avoiding zero. The derivative of $z^{\alpha }$ is bounded on that annulus, proving finite changed exit time from its wedge image. The exponent zero is excluded because the map is constant and its image is not an open domain.