= Solution
Restrict $\mu_x$ to outer boundaries that stay in $D$ and surround $y$. By part (B1), this is the image under outer-boundary formation of rooted Brownian loops through $x$ whose outer boundary stays in $D$ and surrounds $y$. But this restriction is exactly the restriction of $\rho_D$ to loops surrounding both $x$ and $y$. Interchanging $x$ and $y$ gives the same intrinsic restriction of $\rho_D$, so it can equally be represented by outer boundaries of Brownian loops rooted at $y$ that surround $x$.
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