Deligne's purity and compatible-systems conjecture for lisse sheaves

Let $p$ and $\ell$ be distinct primes. Let $X$ be a connected normal scheme of finite type over $\mathbb{F}_p$, and let $\mathcal{E}$ be an irreducible lisse $\overline{\mathbb{Q}}_\ell$-sheaf whose determinant has finite order. For a closed point $x$ of $X$, write $\operatorname{Frob}_x$ for its Frobenius and choose a geometric point $\bar{x}$ above it. Then $\mathcal{E}$ is pure of weight $0$; there is a number field $E\subset\overline{\mathbb{Q}}_\ell$ such that $\det(1-\operatorname{Frob}_xt,\mathcal{E}_{\bar{x}})$ has coefficients in $E$ for every closed point $x$; for every non-archimedean place $\lambda$ of $E$ prime to $p$, its roots are $\lambda$-adic units; and, for sufficiently large $E$, there is for every such $\lambda$ an $E_\lambda$-sheaf $\mathcal{E}_\lambda$ compatible with $\mathcal{E}$, meaning that

$$\det(1-\operatorname{Frob}_xt,\mathcal{E}_{\bar{x}})=\det(1-\operatorname{Frob}_xt,\mathcal{E}_{\lambda,\bar{x}})$$

for every closed point $x$. Deligne's conjecture. These four properties should hold simultaneously. The conjecture is proved for curves by L. Lafforgue; parts concerning purity, integrality, and coefficients are known in general, and the compatible-systems assertion is proved for smooth varieties, while the full normal-variety statement is not established here.