Radial restriction for a conditioned Brownian path ($\mathbb{P}(B\subset U)=|{\Phi }_U^{\prime }(1)|$)

For a Brownian path in the unit disc from $0$ conditioned to exit at $1$, let ${\Phi }_U:U\to D$ fix $0$ and $1$, and suppose $U$ agrees with the disc near $1$. The probability of avoiding $D\setminus U$ is the ratio of the boundary Poisson-kernel densities, hence $|{\Phi }_U^{\prime }(1)|$. Given avoidance, the conformally mapped path has the original law up to the conformal Brownian clock. There is no conformal-radius factor at the interior starting point.