Nilpotent support conjecture for type I graded Lie algebras

Let $G$ be the connected reductive group with involution $\theta$, let $K=G^\theta$, and let $\mathfrak g_1$ be the degree-one component of the associated graded Lie algebra. For a positive integer $a$, let $d=\operatorname{gcd}(a,m)$, let ${}^0_a\Sigma_{m,{\mathbf d}}$ be the specified set of parameters, and let $(\widehat{\mathbb Z/d_\mu\mathbb Z})_a$ denote the characters of order $a$. For an orbit ${\mathcal O}_\mu\subset{\mathcal N}_{\pm1}$ and $\psi_a\in\widehat{\mathbb Z/d_\mu\mathbb Z}$, write ${\mathcal E}_{\psi_a}$ for the corresponding irreducible $K$-equivariant local system. Nilpotent support conjecture.

$$\operatorname{Char}^{\mathrm n}_K({\mathfrak g}_1)_a=\{\operatorname{IC}({\mathcal O}_{\mu},{\mathcal E}_{\psi_a})\mid\mu\in{}^0_a\Sigma_{m,{\mathbf d}},\,\psi_a\in(\widehat{\mathbb Z/d_\mu\mathbb Z})_a\}.$$

This conjecturally describes all character sheaves with nilpotent support in each central-character component for type I graded Lie algebras. The supplied text does not state whether the conjecture is known or remains open.