The scaling-limit and dimension conjecture for three-dimensional loop clusters

Let $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^3}$ be the loop soup on the metric graph of $\mathbb{Z}^3$, and let $\mathcal{L}_{1/2}^{\mathbb{R}^3}$ be the corresponding loop soup in $\mathbb{R}^3$. Scaling-limit and dimension conjecture. The scaling limit of loop clusters of $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^3}$ is exactly the collection of loop clusters of $\mathcal{L}_{1/2}^{\mathbb{R}^3}$. Moreover, the dimension of these clusters equals $\frac{5}{2}$. This conjecture identifies the continuum loop clusters as the scaling limit of the discrete metric-graph clusters and predicts their common dimension; establishing this identification and dimension remains open.