The positive-depth Lusztig induction and Kim–Yu type structure conjecture

Assume $\mathbf{G}'$ is a twisted Levi subgroup of $\mathbf{G}$ containing $\mathbf{T}$. Let $\alpha$ be any representation of $\mathbb{G}'_r(\mathbb{F}_q)$ of depth $<r$, and let $\phi\colon \mathbb{G}'_r(\mathbb{F}_q)\to\overline{\mathbb{Q}}_\ell^{\times}$ be a $(\mathbf{G}',\mathbf{G})$-generic character of depth $r$. Write $R_{\mathbb{G}'_r}^{\mathbb{G}_r}$ for positive-depth Lusztig induction, $K_0\subseteq G_{\mathbf{x},0}$ for the relevant compact subgroup, and let $\rho_{\alpha\otimes\phi}$ be the representation constructed in Yu's construction in the context of Kim–Yu types. Positive-depth Lusztig induction conjecture. One has

$$\left|R_{\mathbb{G}'_r}^{\mathbb{G}_r}(\alpha\otimes\phi)\right|=\operatorname{Ind}_{K_0}^{G_{\mathbf{x},0}}\left(\epsilon^{G/G'}\otimes\rho_{\alpha\otimes\phi}\right),$$

where $\epsilon^{G/G'}$ is the FKS twist. This conjectural identity is intended to relate positive-depth Lusztig induction directly to the construction of Kim–Yu types; it is motivated by Howe factorizations, positive-depth character sheaves, and the known successive-induction structure of the modified Deligne–Lusztig representation, but no resolution is supplied here.