Modularity conjecture for character-sheaf packets

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent in ${\mathscr D}_G(G)$, and let $n_e$ be as in the preceding conjecture. Set

$${\mathcal M}_e={\mathcal M}_e^{perv}[n_e]\subset e{\mathscr D}_G(G).$$

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds; it is rigid with

$$M^*\cong({\mathbb D}^-_G M)[2n_e]$$

for $M\in{\mathcal M}_e$; and it is a modular category, with ribbon structure given by the restriction of the canonical automorphism $\theta$.