The regular Kim–Yu type realization conjecture

Assume that $p\neq 2$ is not bad for $\mathbf{G}$, and let $(\mathbf{T},\theta)$ be an unramified Howe-factorizable pair. Let $\vec\phi=(\phi_{-1},\ldots,\phi_d)$ be a Howe factorization for $\theta$, and let $r_{\mathbb{T}_r}^{\mathbb{G}_r}(\theta)$ denote the modified positive-depth Deligne–Lusztig representation defined from this factorization. Let $G_{\mathbf{x},0}$ be the associated parahoric quotient and $K_0$ the subgroup carrying the associated Kim–Yu type. Regular Kim–Yu type realization conjecture. The representation $r_{\mathbb{T}_r}^{\mathbb{G}_r}(\theta)$ of $G_{\mathbf{x},0}$ is a linear combination of Bushnell–Kutzko types, and its restriction to $K_0$ contains the AFMO-twisted Kim–Yu type associated to $(\mathbf{T},\theta)$. This would provide a general realization of regular Kim–Yu types through modified positive-depth Deligne–Lusztig induction; the statement is presented as a conjecture, with no resolution given here.