Physical rigidity conjecture for Kloosterman sheaves

Let $k$ be the finite base field, let $\check{G}$ be the dual group, and let $\operatorname{Kl}_{\check{G}}(\phi,\chi)$ be the Kloosterman sheaf associated with a generic linear function $\phi$ and a multiplicative character $\chi$. Let $\overline{k}$ be an algebraic closure and set $U={\mathbb P}^1_{\backslash\{0,\infty\}}\otimes_k\overline{k}$. Let $L$ be a $\check{G}$-local system on $U$, and let $\mathscr I_0$ and $\mathscr I_\infty$ denote the inertia groups at $0$ and $\infty$. Physical rigidity conjecture. If the $\mathscr I_0$- and $\mathscr I_\infty$-representations on $L$ have the same isomorphism types as those on $\operatorname{Kl}_{\check{G}}(\phi,\chi)$, then

$$L\cong\operatorname{Kl}_{\check{G}}(\phi,\chi)$$

over $U$. This generalizes the known rigidity phenomenon for the classical $\operatorname{GL}_n$ Kloosterman sheaves.