The tensor-functor realization conjecture for finite-field motives

Let $\mathcal C_{\mathbb F_p}$ be the tensor category of the source, let $\mathcal A'$ be its proposed complex representation-theoretic analogue, and let $({\mathcal A}')^{\mathrm{kar}}$ be the Karoubi closure of $\mathcal A'$. Let $\phi_n$ and $\phi_n^{\mathcal A'}$ be the corresponding realization functors, and let $\operatorname{Fr}_X$ denote the Frobenius element of an object $X$.

Tensor-functor realization conjecture. For every prime $p$ there exists a tensor functor

$$\Phi_p:\mathcal C_{\mathbb F_p}\to({\mathcal A}')^{\mathrm{kar}}$$

and, for every $n\ge 1$, an isomorphism of tensor functors

$$\operatorname{iso}_{n,p}:\phi_n^{\mathcal A'}\circ\Phi_p\simeq i_{\operatorname{Vect}_{\mathbb Q}\to\operatorname{Vect}_{\mathbb C}}\circ\phi_n,$$

where $i_{\operatorname{Vect}_{\mathbb Q}\to\operatorname{Vect}_{\mathbb C}}$ is the scalar-extension embedding. Moreover, for every $X\in\mathcal C_{\mathbb F_p}$, $\Phi_p$ maps $\operatorname{Fr}_X$ to $\operatorname{Fr}_{\Phi_p(X)}$.

This is the paper's Master Conjecture: it seeks a tensor realization of the finite-field motivic category inside a representation-theoretic category while preserving all numerical realizations and Frobenius elements. It is open in the source.