Canonical Kazhdan–Lusztig cell conjecture on completely prime primitive ideals

Let $W_a$ be the affine Weyl group, let ${^L \mathfrak{g}}$ be the Langlands-dual Lie algebra with Cartan subalgebra ${^L \mathfrak{h}}$, and let $\mathcal{C}$ be a canonical left Kazhdan–Lusztig cell in $W_a$. Write $\bar{\mathcal C}$ for its associated closed region and identify ${^L \mathfrak{h}}^*$ with $\mathfrak{h}$. Let $\lambda \in {^L \mathfrak{h}}^* \cong \mathfrak{h}$ be an element of minimal length in $\bar{\mathcal C}$.

Canonical cell conjecture. The element of minimal length is unique, and $J(\lambda)$ is a completely prime maximal ideal of ${\mathbf U}({^L \mathfrak{g}})$ whose associated variety is the orbit indexing $\mathcal{C}$.

This conjecture connects canonical left Kazhdan–Lusztig cells with completely prime maximal ideals and their associated nilpotent orbits. The rank-two examples in types $B_2$ and $G_2$ motivate the claim, while the general assertion is presented as a conjecture.