Deligne's companions conjecture for normal varieties

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $X_0$ be a connected normal variety over $\mathbb{F}_q$, let $\ell\ne p$ be a prime, and let $\mathcal{V}_0$ be an irreducible Weil lisse $\overline{\mathbb{Q}}_{\ell}$-sheaf over $X_0$ with finite-order determinant. Let $E$ be the number field generated by the coefficients of the Frobenius polynomials of $\mathcal{V}_0$ at closed points. A Weil lisse $E_{\lambda}$-sheaf is $E$-compatible with $\mathcal{V}_0$ when it has the same Frobenius polynomials under the embedding determined by the finite place $\lambda$. Companions conjecture. After possibly replacing $E$ with a finite extension, for every finite place $\lambda$ not dividing $p$ there exists a Weil lisse $E_{\lambda}$-sheaf $E$-compatible with $\mathcal{V}_0$. This conjecture extends the known companion results for smooth varieties to normal varieties; the paper studies the obstruction in the singular case and proves it for some singular normal varieties. The status of the conjecture in general is open.