Global Kottwitz cohomology conjecture

Let $\operatorname{Kt}_F$ be Kottwitz's $F$-linear tensor category for a local or global field $F$, and let there be natural functors

$$\operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb Q_p},\qquad \operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb Q_\ell},\qquad \operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb R}.$$

For a variety $X$ over $\overline{\mathbb F}_p$, global Kottwitz cohomology conjecture. There should be a Weil cohomology theory $H^i_{\operatorname{Kt}_{\mathbb Q}}(X)$ with values in $\operatorname{Kt}_{\mathbb Q}$ which maps to crystalline cohomology under $\operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb Q_p}$, maps to étale cohomology under $\operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb Q_\ell}$ for $\ell\ne p$, via the fully faithful embedding of finite-dimensional $\mathbb Q_\ell$-vector spaces, and restricts under $\operatorname{Kt}_{\mathbb Q}\to\operatorname{Kt}_{\mathbb R}$ to a Weil cohomology theory $H^i_{\operatorname{Kt}_{\mathbb R}}(X)$ with values in $\operatorname{Kt}_{\mathbb R}$. This would provide a linear-algebraic global object simultaneously encoding the crystalline, étale, and real realizations; the source presents it as an important open problem and gives no known construction.