Four- and five-dimensional Brownian loop-soup gluing conjecture

For $d\geq4$, consider the Brownian loop-soup in $(0,1)^d$ and the operation of contracting each loop, so that percolation can be viewed as a mechanism for gluing loops into clusters. Brownian loop-soup gluing conjecture. When $d=5$, critical percolation in the contracted-loop space should exist and be non-trivial; equivalently, observing the Brownian loop-soup alone does not determine which loops belong to the same clusters. When $d=4$, the gluing mechanism should be deterministic; equivalently, the Brownian loop-soup alone determines which loops belong to the same clusters. The conjecture is motivated by the analogy with CLE clustering: the five-dimensional case is expected to retain non-trivial randomness, while the four-dimensional case is expected to have a deterministic gluing rule.