We consider the Schramm-Loewner evolution ( ${\mathrm{SLE}}_{\kappa }$ ) for $\kappa \in (4,8)$ , which is the regime that the curve is self-intersecting but not space-filling. We let $\mathcal{K}$ be the set of $\kappa \in (4,8)$ for which the adjacency graph of connected components of the complement of an ${\mathrm{SLE}}_{\kappa }$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_{1},\ldots ,U_{n}$ with $U_{1} = U$ , $U_{n} = V$ , and $\partial U_{i} \cap \partial U_{i+1} \neq \emptyset $ for each $1 \leq i \leq n-1$ . It was proved by Gwynne and Pfeffer [11] that this set is non-empty. We show that the range of an ${\mathrm{SLE}}_{\kappa }$ for $\kappa \in \mathcal{K}$ is a.s. conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an ${\mathrm{SLE}}_{\kappa }$ for $\kappa \in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.
