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

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 $\gamma \neq 0$. 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$_{\kappa}$, $\kappa \in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.