The intermediate-variety dimension conjecture for orbit closures

Let ${\cal K}$ and ${\cal H}_{\overline{y_e}}$ be the stabilizer-related groups associated with the orbit-closure data, and let $\overline{O(z)}\subseteq\overline{O(y)}$ be orbit closures. An intermediate variety $W$ is a $G$-stable closed variety satisfying $\overline{O(y)}\supseteq W\supseteq\overline{O(z)}$, and it is strict when both inclusions are strict. Intermediate-variety conjecture. If

$$\dim({\cal K})<\dim({\cal H}_{\overline{y_e}}),$$

then there is a strictly intermediate variety

$$\overline{O(z)}\subsetneq W\subsetneq\overline{O(y)}$$

of dimension $\dim(G)-\dim({\cal H}_{\overline{y_e}})$. The claim predicts an intermediate geometric object in the normal-cone construction when the relevant stabilizer dimensions differ; the supplied text does not state whether this conjectural formulation has been resolved.