 Solution

Restrict $\mu$ first to loops surrounding $0$. This satisfies the pointed restriction property, so for one constant $C$,A single rooted pushforward cannot equal the whole ${\mu }_D$: it only sees loops surrounding its root. The rest is recovered by moving that root.

For $z\in D$, choose a disc automorphism ${\psi }_z$ with ${\psi }_z(0)=z$. Conformal restriction yieldswith the same constant. Let $(z_j)$ be a countable dense set in $D$. Every simple loop contained in $D$ surrounds one of these points. Make this cover disjoint by $P_j=E_{z_j}\setminus {\bigcup }_{i<j}{E}_{z_i}$. For a measurable event $A$ of loops in $D$,This explicitly relates the entire loop measure to the single-root measure without overcounting a loop once for every point in its interior. It also proves that knowledge of $\rho$, up to normalization, determines ${\mu }_D$. Nontriviality gives $C>0$: otherwise all countably many pointed patches would vanish. Using the usual unrooted Brownian loop measure, the same construction is commonly expressed as its outer-boundary pushforward, up to a constant.