Compact Fourier transform compatibility for pure inner twists

Let $G$ be a connected reductive $F$-split group. For a unipotent element $u otin G^\text{?}$, a compact pair $(s,h) otin \text{?}$, and a maximal compact open subgroup $K$ of $G$, let $\operatorname{res}_K$ denote parahoric restriction and let ${\rm FT}_K$ denote the corresponding nonabelian Fourier transform. Compact Fourier transform compatibility conjecture. There \exists a root of unity $\text{?}$ such that

$$\operatorname{res}_K(\text{?})=\text{?}({\rm FT}_K\text{?})(\text{?}).$$

This predicts compatibility between the dual compact nonabelian Fourier transform on compact representations and the finite reductive-group Fourier transform after parahoric restriction. The supplied text gives no resolution status.