Cuspidal character sheaves from local systems for stably graded Lie algebras

Let ${\mathfrak{g}}_1$ be the degree-one component of one of the stably graded Lie algebras considered here, let $K$ act on ${\mathfrak{g}}_1$, and let ${\mathfrak{g}}_1^{rs}$ be its regular-semisimple locus. Define

$$\Theta_{({\mathfrak{g}}_1,K)}=\{\text{irreducible representations of }\pi_1^K({\mathfrak{g}}_1^{rs})\text{ that appear as composition factors of }{\mathcal M}_\chi,\ \chi\in\hat I\}.$$

For each $\pi\in\Theta_{({\mathfrak{g}}_1,K)}$, let ${\mathcal L}_\pi$ be the corresponding $K$-equivariant local system on ${\mathfrak{g}}_1^{rs}$. Cuspidal character-sheaf conjecture. The set of cuspidal character sheaves on ${\mathfrak{g}}_1$ is precisely

$$\left\{\operatorname{IC}({\mathfrak{g}}_1^{rs},{\mathcal L}_\pi)\mid\pi\in\Theta_{({\mathfrak{g}}_1,K)}\right\}.$$

This asserts that, for the stable gradings under consideration, every cuspidal character sheaf is obtained from one of these regular-semisimple local systems and that all members of the displayed set are cuspidal. The claim gives a proposed classification of the cuspidal part of the character-sheaf theory in this setting; no resolution is supplied here.