Char $p$ Mumford–Tate conjecture for ordinary abelian schemes

Let $X_0$ be a geometric connected smooth variety over $\mathbb{F}_q$, let $f_0:X_0\rightarrow \mathscr{A}_{g,\mathbb{F}_q}^{{\mathrm{ord}}}$ be a morphism as above, and let $A$ be the pullback abelian scheme over $X_0$. Denote by $G_{\mathrm{B}}(A)$ the generic Mumford–Tate group of the smallest Shimura subvariety whose Zariski closure contains the image of $f_0$. For $l\neq p$, let $G_l(A)$ be the $l$-adic étale monodromy group, and let $G_p(A)$ be the $p$-adic overconvergent crystalline monodromy group. Char $p$ Mumford–Tate conjecture. One has

$$G_l(A)^\circ=G_{\mathrm{B}}(A)_{\mathbb{Q}_l}$$

for every $l\neq p$, and

$$G_p(A)^\circ=G_{\mathrm{B}}(A)_{\mathbb{Q}_p}.$$

This conjecture predicts that the connected $l$-adic and $p$-adic monodromy groups of an ordinary abelian scheme in characteristic $p$ coincide with the corresponding realizations of its generic Mumford–Tate group. It is presented as a monodromy conjecture in the paper; its resolution is not specified in the supplied text.