The critical connectivity and Sierpiński-carpet conjecture for SLE

For $\kappa \in (4,8)$, let $\eta$ be an SLE$_\kappa$ curve from $0$ to $\infty$ in $\mathbb{H}$. The adjacency graph of the connected components of $\mathbb{H} \setminus \eta$ has vertices given by those components, with two vertices adjacent when the corresponding boundaries intersect. Critical SLE connectivity conjecture. There exists $\kappa_c \in (4,8)$ such that for $\kappa \in (4,\kappa_c)$ this graph is almost surely connected, whereas for $\kappa \in (\kappa_c,8)$ it is almost surely disconnected and the range of $\eta$ contains a topological Sierpiński carpet. This conjecture proposes a phase transition between connected and disconnected bubble adjacency, with the latter accompanied by Sierpiński-carpet structure; the critical value and behavior at $\kappa=\kappa_c$ remain open.