Crew's petites camarades conjecture for curves

Let $k$ be a finite field of characteristic $p$, let $U$ be a smooth geometrically connected curve over $k$, and let $\mathscr{F}$ be an irreducible smooth $\overline{\mathbb{Q}}_\ell$-sheaf on $U$ whose determinant is defined by a finite-order character of $\pi_1(U)$. Crew's petites camarades conjecture. There exists a number field $E$ such that, for every $x\in|U|$, the characteristic polynomial of geometric Frobenius acting on $\mathscr{F}_{\overline{x}}$ has coefficients in $E$, and, for every place $\mathscr{P}$ of $E$ dividing $p$, there is an overconvergent $F$-isocrystal on $U/E_{\mathscr{P}}$ whose Frobenius characteristic polynomial at every $x\in|U|$ agrees with that of $\mathscr{F}$. This is a $p$-adic companion statement for irreducible $\ell$-adic sheaves with finite-order determinant. The text presents it as a formulation of Deligne's earlier petites camarades conjecture; the supplied status is unknown, so it is recorded as open.