Frenkel–Gross comparison conjecture for Kloosterman connections

Let $k={\mathbb C}$, let $G$ be almost simple, and let $\mathcal G$ be a quasi-split form of $G$ on ${\mathbb P}^1_{\backslash\{0,\infty\}}$ determined by $\sigma:\mu_N\to\operatorname{Aut}^\dagger(G)$. Let $I(1)/I(2)$ be the relevant quotient of the Iwahori subgroup, let $\phi:I(1)/I(2)\to{\mathbb G}_a$ be a generic linear function, and let $({\check X}_0,\ldots,{\check X}_{r_\sigma})$ be the corresponding basis of the indicated negative-root spaces for the twisted affine Kac–Moody algebra. Frenkel–Gross comparison conjecture. There is a bijection between generic linear functions $\phi$ and such bases, and corresponding pairs satisfy a natural isomorphism

$$\operatorname{Kl}_{{}^L\!\mathcal G}(\phi)_{\operatorname{dR}}\cong({}^L\!\mathcal G,\nabla_{{}^L\!\mathcal G}({\check X}_0,\ldots,{\check X}_{r_\sigma})).$$

This predicts that the paper's de Rham construction agrees with the Frenkel–Gross connection.