Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:A \to S_0$ be an abelian scheme of relative dimension $g$ such that the Newton polygon of $A[p^{\infty}]$ is constant. For all $\ell \not=p$ we can consider the local systems $R^1 \pi_{\ast} \mathbb{Q}_{\ell}$ as a representation $\rho_{\ell}$ of $\pi_1^{\text{ét}}(S)$, and deep results of Deligne tell us that this is a semisimple representation; in fact the Zariski closure of the image of $\rho$ is "independent of $\ell$". Similarly there is a a $p$-adic variant $\mathcal{E}$ of $R^1 \pi_{\ast} \mathbb{Q}_{\ell}$, which is an (overconvergent) isocrystal, it also has the same monodromy group (see https://arxiv.org/abs/1711.06669) as its $\ell$-adic cousins.

Let $\mathbb{X}_b$ be a $p$-divisible group over $\overline{\mathbb{F}}_p$ with the same Newton polygon as $A[p^{\infty}]$ and let $J_b(\mathbb{Q}_p) \subset G(\breve{\mathbb{Q}}_p)$ be the automorphism group of the isocrystal associated to $\mathbb{X}_b$. Then Proposition 4.3.13 of Caraini-Scholze (https://arxiv.org/abs/1511.02418) gives us a pro-étale $J_b(\mathbb{Q}_p$)-torsor which roughly speaking parametrises quasi-isogenies $A[p^{\infty}] \to \mathbb{X}_b$.

**Question**: Is there any relation between the Zariski closure of the image of $\rho':\pi_1^{\text{ét}}(S) \to J_b(\mathbb{Q}_p)$ and the geometric monodromy groups of $\rho_{\ell}$ and $\mathcal{E}$?

**Question**: Is the Zariski closure of the image of $\rho':\pi_1^{\text{ét}}(S) \to J_b(\mathbb{Q}_p)$ reductive?

**Example**: If $S_0=Y_1(N)^{\text{ord}}$ is the mod $p$ fiber of the ordinary locus of the modular curve, then the $\ell$-adic monodromy group is equal to $\operatorname{SL}_2$ and the Zariski closure of the image of $\rho'$ is $\mathbb{G}_m=J_b$.