Sommers's cell-region decomposition conjecture

Let $\mathcal{C}$ be the canonical left Kazhdan–Lusztig cell indexed by a nilpotent orbit ${^L \mathcal O} \subset {^L \mathfrak{g}}$. For each pair $(\mathcal O,C)$ consisting of a nilpotent orbit $\mathcal O$ and a conjugacy class $C$ in its component group, let $\bar{\mathcal D}_{(\mathcal O,C)}$ denote the associated canonical left $\tilde{S}$-region, and let $d_{(\mathcal O,C)}$ be the map to nilpotent orbits of ${^L \mathfrak{g}}$.

Sommers's cell-region decomposition conjecture.

$$\bar{\mathcal C}=\bigcup_{d_{(\mathcal O,C)}={^L \mathcal O}}\bar{\mathcal D}_{(\mathcal O,C)}.$$

This conjecture would explain how Kazhdan–Lusztig cell regions decompose into canonical left $\tilde{S}$-regions and would imply that the elements supplied by the first conjecture are among those supplied by the second. It is attributed in the paper to the second author and remains open here.