The generalized Tate conjecture

Let $X$ be a smooth projective variety, or a smooth and proper Deligne--Mumford stack with projective coarse moduli space, over a finitely generated field $L$. Let $L^{\mathrm{s}}$ be a separable closure of $L$, and let $V\subset H^j(X_{L^{\mathrm{s}}},\mathbb{Q}_\ell)$ be a Galois subrepresentation. Suppose that $V(k)$ is effective, meaning that the eigenvalues of Frobenius elements acting on $V(k)$ at good places are algebraic integers.

Generalized Tate conjecture. There is a closed algebraic subset $Z\subset X$ of codimension $k$ such that

$$V\subset\ker\bigl(H^j(X_{L^{\mathrm{s}}},\mathbb{Q}_\ell)\to H^j((X\smallsetminus Z)_{L^{\mathrm{s}}},\mathbb{Q}_\ell)\bigr).$$

This is the arithmetic analogue of the generalized Hodge conjecture, replacing Hodge coniveau by effectivity of the twisted Galois representation. It remains open in general.