The Lusztig induction and dual restriction conjecture

Let $H$ be the reductive group and let $\widehat{P}\subseteq\widehat{H}$ be the parabolic data defining the Lusztig induction functor

$$\operatorname{Ind}_{\widehat{P}}^{\widehat{H}}: \operatorname{SPer}(H_{\lambda_-^{\mathbb{Q}_p}} \backslash V_{\lambda_-^{\mathbb{Q}_p}}) \longrightarrow \operatorname{SPer}(H_{\lambda^{\mathbb{Q}_p}} \backslash V_{\lambda^{\mathbb{Q}_p}}).$$

On the level of Grothendieck groups, the dual restriction conjecture. one has

$$\mathcal{D}_{\tilde{\lambda}_j}^* = \operatorname{Ind}_{\widehat{P}}^{\widehat{H}}.$$

This identifies the dual operation $\mathcal{D}_{\tilde{\lambda}_j}^*$ with Lusztig induction; the source states that a proof is work in progress.