Lusztig's extension-by-zero conjecture for the semisimple character sheaf

Let $G^1$ be the relevant component, let $\overline{G^1}^{ss}$ be its semistable locus, and let $S'$ be the simple perverse sheaf on $\overline{G^1}^{ss}$ obtained from the semisimple character sheaf. Let $IC_{\overline{G^1}}(S')$ denote its intermediate extension to $\overline{G^1}$. Extension-by-zero conjecture. The intermediate extension of $S'$ to $\overline{G^1}$ is the extension by $0$ outside $\overline{G^1}^{ss}$. The paper proves the equivalent characterization of the semistable locus through the stalks of the corresponding simple perverse sheaf, thereby reducing Lusztig's conjecture to this extension statement; the remaining assertion is not resolved in the source.