Strong rigidity characterization conjecture for Kloosterman sheaves

Let $k$ be the finite base field, let $\check{G}$ be the dual group with maximal torus $\check{T}$, let $T$ be the corresponding torus, and let $U={\mathbb P}^1_{\backslash\{0,\infty\}}$. Suppose $L$ is a $\check{G}$-local system on $U$ such that $L$ is tame at $0$, the semisimple part of a topological generator of tame inertia at $0$ is conjugate to an element of $\check{T}[q-1]$ corresponding to a multiplicative character $\chi:T(k)\to\overline{{\mathbb Q}}_\ell^\times$, and

$$\operatorname{Swan}_\infty(L^{\operatorname{Ad}})=r,\qquad (L^{\operatorname{Ad}})^{\mathscr I_\infty}=0.$$

Strong rigidity characterization conjecture. There exists a generic linear function $\phi:I(1)/I(2)\to{\mathbb G}_a$ such that $L\cong\operatorname{Kl}_{\check{G}}(\phi,\chi)$ up to an unramified twist given by a homomorphism $\operatorname{Gal}(\overline{k}/k)\to Z\check{G}$. This would characterize Kloosterman sheaves from their local ramification data, up to the stated central twist.