Tate conjecture in characteristic p for integral Shimura models

Let $k(v)$ be the residue field at $v$, let $\kappa$ be a perfect field over $k(v)$, and let $x\in \mathscr{S}_K(\kappa)$. For each prime $\ell$, write $I_{\ell,x}$ for the corresponding realization group, and let $\operatorname{Aut}^p(x)$ denote the prime-to-$p$ automorphism group appearing in the realization construction. Tate conjecture in characteristic $p$. There exists a reductive group $I_x$ over $\mathbb{Q}$ with realization maps

$$\alpha_\ell\colon I_x\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\to I_{\ell,x}$$

for all primes $\ell$, satisfying

$$I_x(\mathbb{Z}_{(p)})=I_x(\mathbb{Q})\cap I_{\mathbb{Z}_p,x}(\mathbb{Z}_p)=\operatorname{Aut}^p(x),$$

and such that the map $I_x(\mathbb{Z}_{(p)})\to I_{\ell,x}(\mathbb{Q}_\ell)$ is the realization map from the cited quasi-isogeny construction. If $\kappa$ is finite, then for every prime $\ell$ the realization map

$$I_x\otimes_{\mathbb{Q}}\mathbb{Q}_\ell\to I_{\ell,x}$$

is an isomorphism. This is the characteristic-$p$ analogue of the preceding Tate conjecture, combining the integral prime-to-$p$ realization property with the asserted isomorphism over finite fields. The source provides no evidence resolving it.