Inversion invariance conjecture for full-plane CLE6 and k-arm IIC limits

Let $\mathbb{C}$ be the full plane, let $\mu_k$ denote the scaling limit of the $k$-arm IIC for $k\in\{1,2,4\}$, and consider inversion $z\mapsto 1/z$. Inversion-invariance conjecture. The full-plane $\operatorname{CLE}_6$ and each $\mu_k$ for $k\in\{1,2,4\}$ are invariant under $z\mapsto 1/z$. The source proves existence of these scaling limits but says that inversion invariance is not established by the cited construction; the conjecture proposes this additional symmetry.