Uniqueness conjecture for the Brownian path decomposition

Let $D$ be the unit ball, let $(\mathcal K_D,d_H)$ be the space of compact subsets of $D$ equipped with the Hausdorff metric, and let $\Gamma$ be the set of simple paths starting at $0$ and meeting $\partial D$ only at their endpoint. Let $\mathrm{K}_1$ and $\mathrm{K}_2$ be random elements of $(\mathcal K_D,d_H)$ such that $\mathrm{K}_1,\mathrm{K}_2\in\Gamma$ almost surely, and let $\mathrm{BS}$ be a Brownian loop soup in the unit ball independent of both.

Uniqueness conjecture. If, for each $i\in\{1,2\}$, the union of $\mathrm{K}_i$ and all loops from $\mathrm{BS}$ intersecting $\mathrm{K}_i$ has the same law as the trace of Brownian motion stopped upon exiting the unit ball, then $\mathrm{K}_1$ and $\mathrm{K}_2$ have the same law in $(\mathcal K_D,d_H)$.

In two dimensions, the analogous decomposition is obtained from the scaling limit of loop-erased random walk, namely $\mathrm{SLE}_2$; in three dimensions the paper proves that subsequential scaling limits are simple paths and gives the corresponding Brownian-loop-soup decomposition, but uniqueness of the law of the simple path remains conjectural.