Braverman–Kazhdan's exactness and induction conjecture for gamma sheaves

Let $G$) be a reductive group over an algebraically closed field of finite characteristic, let $\breve G$ be its dual group, and let $\rho:\breve G\rightarrow\operatorname{GL}(V_\rho)$ be a representation satisfying the technical conditions in the source. Let $\Psi_{G,\rho}$ be the associated gamma sheaf and define

$$\mathrm F_{G,\rho}:=(-)*\Psi_{G,\rho}:D^b_c(G,\bar{\mathbb Q}_\ell)\rightarrow D^b_c(G,\bar{\mathbb Q}_\ell).$$

Braverman–Kazhdan's conjecture. The functor $\mathrm F_{G,\rho}$ is exact with respect to the perverse $t$-structure, and it commutes with induction functors. This conjecture concerns the non-linear analogue of the Fourier–Deligne transform. The source states these properties as the conjectured parallels of exactness and compatibility with induction for the ordinary Fourier–Deligne transform; the paper proves the corresponding acyclicity result in the de Rham setting and derives the induction property, while exactness is the remaining assertion in this formulation.