Codimension-one connectedness conjecture for associated graphs

Let $X$ be an irreducible Harish-Chandra $(\mathfrak{g},K)$-module, and let

$$\operatorname{\mathcal{AV}}(X)=\bigcup_{i=1}^{\ell}\overline{\mathbb{O}^K_i}$$

be the irreducible decomposition of its associated variety. The graph $\operatorname{\mathcal{AV}^\Gamma}(X)$ has these irreducible components as vertices, with edges inherited from the orbit graph: two vertices are joined when the intersection of their closures contains a nilpotent $K$-orbit of codimension one. Codimension-one connectedness conjecture. The graph $\operatorname{\mathcal{AV}^\Gamma}(X)$ is connected. Equivalently, for every pair $(\mathbb{O}^K_i,\mathbb{O}^K_j)$, there is a sequence

$$\mathbb{O}^K_i=\mathbb{O}^K_{i_0},\;\mathbb{O}^K_{i_1},\;\mathbb{O}^K_{i_2},\;\dots,\;\mathbb{O}^K_{i_n}=\mathbb{O}^K_j$$

such that, for $1\leq k\leq n$, $\overline{\mathbb{O}^K_{i_{k-1}}}\cap\overline{\mathbb{O}^K_{i_k}}$ contains a nilpotent $K$-orbit of codimension one. The conjecture concerns the codimension-one connectedness of irreducible components of associated varieties of Harish-Chandra modules. Vogan's theorem shows that the boundary of each component has codimension one when the associated variety is reducible, but the boundary may have components that do not connect distinct $K$-orbits. The general connectedness assertion remains open, and the source allows $K$ to be disconnected.