Braverman–Kazhdan's kernel conjecture for rho-Bessel sheaves

Let $G$ be a reductive group over $\mathbb F_q$, let $\rho$ be a representation as in the Braverman–Kazhdan construction, and let $\Phi_{G,\rho,\psi}$ be the corresponding $\rho$-Bessel sheaf equipped with a Weil structure. Let $\phi_{G,\rho,\psi}:G(\mathbb F_q)\to\overline{\mathbb Q}_\ell$ be the non-linear Fourier kernel, and let ${\operatorname{Tr}}(\Phi_{G,\rho,\psi})$ denote the function associated to the sheaf by the functions–sheaves correspondence. Braverman–Kazhdan's kernel conjecture. One has

$${\operatorname{Tr}}(\Phi_{G,\rho,\psi})=\phi_{G,\rho,\psi}.$$

The conjecture gives a geometric construction of the non-linear Fourier kernel. The paper states that it was proved in the cases of semisimple rank at most one and $G={\operatorname{GL}}_n$ under an assumption on $\rho$, and later by Laumon and Letellier without such an assumption.