Kloosterman-sheaf rigidity conjecture for prime rank

Let $n$ be prime and let $\psi:\mathbb F_q\to\overline{\mathbb Q}_\ell^{\times}$ be nontrivial. Let $\Gamma=\varinjlim\widehat{\mathbb F_{q^n}^{\times}}$, and let $\xi_i,\eta_i\in\Gamma$ for $1\le i\le n$. Suppose the Kloosterman sheaves $\operatorname{Kl}(\psi;\xi_1,\ldots,\xi_n)$ and $\operatorname{Kl}(\psi;\eta_1,\ldots,\eta_n)$ descend to local systems $\mathcal K_1,\mathcal K_2$ over $\mathbb G_{m,\mathbb F_q}$, equivalently the associated divisors are Frobenius invariant. Kloosterman-sheaf rigidity conjecture. If $\mathcal K_1$ and $\mathcal K_2$ have the same Frobenius traces on every $\mathbb F_q$-point of $\mathbb G_m$, then

$$\mathcal K_1\cong\mathcal K_2,$$

and in particular

$$\sum_{i=1}^n\xi_i=\sum_{i=1}^n\eta_i$$

as divisors on $\Gamma$. The conjecture proposes that pointwise trace data determine these descended Kloosterman local systems and their character divisors; the source gives no resolution.