The semisimplicity conjecture for Galois actions on cohomology

Let $Y$ be a smooth projective variety, or a smooth and proper Deligne--Mumford stack with projective coarse moduli space, over a finitely generated field $L$, and let $L^{\mathrm{s}}$ be a separable closure of $L$. For a prime $\ell$ invertible in $L$, consider the $\ell$-adic cohomology representation of $\operatorname{Gal}(L^{\mathrm{s}}/L)$ on $H^*(Y_{L^{\mathrm{s}}},\mathbb{Q}_\ell)$.

Semisimplicity conjecture. The Galois action on

$$H^*(Y_{L^{\mathrm{s}}},\mathbb{Q}_\ell)$$

is semisimple.

This is closely linked to the Tate conjecture. In characteristic zero, the Tate conjecture implies semisimplicity, but the conjecture remains open in general.