Topological non-equivalence conjecture for independent dense-phase embeddings

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathbb{X}'$ be an independent copy of $\mathbb{X}$. A neighborhood of $\mathbb{X}$ means a neighborhood of this embedded subset in $\mathbb{S}^2$. Topological non-equivalence conjecture. Almost surely, no two neighborhoods of $\mathbb{X}$ and $\mathbb{X}'$ can be mapped to each other by a homeomorphism of $\mathbb{S}^2$. This conjecture expresses the expected randomness of the topology of the embedding in the dense case and is motivated by analogous questions for SLE random fractals and planar Brownian motion; the source gives a heuristic argument but no proof or resolution.