Component-density variant of the crystalline Chebotarëv conjecture

Let ${\mathcal{F}}$ be a convergent $F$-isocrystal on the curve $U$, fix $u\in U({\mathbb{F}}_{q^e})$, and let $\operatorname{Gr}({\mathcal{F}}/U,u)$ be its monodromy group over $K_e$. For each closed point $x\in|U|$, let $\operatorname{Frob}_x({\mathcal{F}})$ be the corresponding Frobenius conjugacy class. Component-density variant. For every subset $S\subset|U|$ of positive upper Dirichlet density, the Zariski closure of

$$\bigcup_{x\in S}\operatorname{Frob}_x({\mathcal{F}})$$

contains a connected component of $\operatorname{Gr}({\mathcal{F}}/U,u)\times_{K_e}{\overline{K}}$. This variant weakens full Zariski density to density in at least one connected component and is presented as a variant of the preceding conjecture. Its resolution is not specified in the supplied text.