Explicit boundary-value formula for character sheaves on the semistable locus

Let $A'$ be a character sheaf on $G^1$, let $I$ be the set indexing the relevant standard Levi subgroups, and let $J\subset I$. Write $Z_{J,\delta;1}$ for the corresponding boundary piece, $L_{J_{\delta}}g_0$ for the associated space, $c_J$ for the operation retaining the summands equivariant under the right $Z(L_J)$-action, and $i^G_{L_{J_{\delta}}}$ for the induction functor. Boundary-value conjecture. For every $J\subset I$,

$$IC(\overline{G^1}, A') \mid_{Z_{J,\delta;1}}=i^G_{L_{J_{\delta}}}(C)[|I-J|],$$

where $C$ is the semisimple perverse sheaf on $L_{J_{\delta}}g_0/Z(L_J)$ whose pullback to $L_{J_{\delta}}g_0$ is

$$c_J\operatorname{res}^{L_{J_{\delta}}g_0}_{G^1}(A')[-|I-J|].$$

The formula gives an explicit description of the boundary restrictions of intermediate extensions of character sheaves and is presented as a conjectural refinement of the preceding theorem and corollary; its resolution status is not specified in the source.