The strong density conjecture for arithmetic representations

Let $X_0$ be a smooth geometrically connected variety over a finite field $k={\mathbb F}_q$ of characteristic $p\neq\ell$, let $X=X_0\otimes_k\bar k$, and let $G=\pi_1^{\rm \acute{e}t}(X,x)$ for a geometric point $x\in X_0(\bar k)$. Fix a prime $\ell\neq p$, a finite field $\mathbb F$ of characteristic $\ell$, and a continuous semisimple representation $\bar\rho:G\to\operatorname{GL}_r(\mathbb F)$. Let $\mathcal S_{\bar\rho}$ be the corresponding representation space with its Zariski topology, let $\Phi$ be the Frobenius automorphism of $\mathcal S_{\bar\rho}$, and let $\mathcal A_{\bar\rho}=\bigcup_{n>0}\mathcal S_{\bar\rho}^{\Phi^n}$ be the set of arithmetic points. If $Z\subset\mathcal S_{\bar\rho}$ is Zariski closed and $\Phi^n(Z)=Z$ for some integer $n>0$, then Strong conjecture. $Z$ is the Zariski closure of its arithmetic points $Z\cap\mathcal A_{\bar\rho}$. This strengthens the weak density assertion by imposing the same density conclusion on every Frobenius-invariant closed subspace; the supplied text gives no resolution status.