Connectivity of the adjacency graph of complementary components of the SLE fan

Suppose that $h$ is an instance of the Gaussian free field (GFF) on a simply connected domain $D\subseteq \mathbf{C}$ and $x,y\in \mathrm{\partial }D$ are distinct. Fix $\kappa \in (0,4)$ and for each $\theta \in \mathbf{R}$ let ${\eta }_{\theta }$ be the flow line of $h$ from $x$ to $y$. Recall that for ${\theta }_1<{\theta }_2$ the fan $\mathbf{F}({\theta }_1,{\theta }_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of ${\eta }_{\theta }$ as $\theta$ varies in any fixed countable dense subset of $[{\theta }_1,{\theta }_2]$. We show that the adjacency graph of components of $D\setminus \mathbf{F}({\theta }_1,{\theta }_2)$ is a.s. connected, meaning it a.s. holds that for every pair $U,V$ of components there exist components $U_1,\dots ,U_n$ so that $U_1=U$, $U_n=V$, and $\mathrm{\partial }{U}_i\cap \mathrm{\partial }{U}_{i+1}\mathrm{\ne }\mathrm{\varnothing }$ for each $1\le i\le n-1$. We further show that $\mathbf{F}({\theta }_1,{\theta }_2)$ a.s. determines the flow lines used in its construction. That is, for each $\theta \in [{\theta }_1,{\theta }_2]$ we prove that ${\eta }_{\theta }$ is a.s. determined by $\mathbf{F}({\theta }_1,{\theta }_2)$ as a set.