Thick points of the planar GFF are totally disconnected for all γ≠0

We prove that the set of γ-thick points a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all γ≠0. Our proof relies on coupling between GFF and nested CLE4. In particular, we show thick are same as those weighted CLE4 nesting introduced in [24] establish almost sure total disconnectedness complement CLEκ, κ∈(8∕3,4]. As corollary see singular supercritical LQG metrics disconnected.