Katz's local decomposition conjecture for arithmetic families on curves

Let $Y$ be a scheme over $\mathbb Z$, and let $\{(Y_p,{\mathscr G}_p)\}$ be a family arising from a fixed constructible complex and morphisms as in the setup. Suppose $Y$ is an algebraic curve over $\mathbb Z$, and let $\overline{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Katz's local decomposition conjecture. For sufficiently large $p$, the inverse image of ${\mathscr G}_p$ under any dominant morphism

$$\operatorname{Spec}\overline{\mathbb F}_p((t))\to Y_p$$

can be written as a direct sum of objects of the form

$$[r]_\ast\bigl({\mathscr L}_{\psi_p}(\alpha(t))\otimes{\mathscr K}\bigr),$$

where $r$ is relatively prime to $p$, $\alpha(t)\in\overline{\mathbb F}_p((t))$, and ${\mathscr K}$ is a tame $\overline{\mathbb Q}_\ell$-sheaf on $\operatorname{Spec}\overline{\mathbb F}_p((t))$. This is a proposed uniform local description of the sheaves in arithmetic families; the supplied source states it as a conjecture and gives no evidence of a resolution.