Thick points of the planar GFF are totally disconnected for all $\gamma \mathrm{\ne }0$

We prove that the set of $\gamma$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $\gamma \mathrm{\ne }0$. Our proof relies on the coupling between a GFF and the nested CLE${}_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE${}_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE${}_{\kappa }$, $\kappa \in (8\mathrm{/}3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.