Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

We construct the canonical geodesic metric on the gasket of conformal loop ensembles (CLE$_κ$) in the regime $κ\in (4,8)$ where the loops intersect themselves, each other, and the domain boundary. Previous work of the authors and V. Ambrosio showed that the subsequential limits associated with certain approximation procedures for such a metric exist and are non-trivial. In this work, we show that the limit exists by proving that there is at most one geodesic metric on the CLE$_κ$ gasket which satisfies certain properties. Further, we obtain that the limit is conformally covariant. This paper is the foundation of future work which show that the metric for $κ=6$ is the continuum scaling limit of the chemical distance metric for critical percolation in two dimensions. We further conjecture that for $κ\in (4,8)$, the geodesic CLE$_κ$ metric is the scaling limit of the chemical distance metric associated with discrete models that converge to CLE$_κ$.