Folklore compatible-systems conjecture for de Rham lisse sheaves

Let $\ell$ be a rational prime. Let $X$ be an irreducible regular scheme, flat and of finite type over $\mathbb{Z}[\ell^{-1}]$. Let $E$ be a finite extension of $\mathbb{Q}$, let $\lambda$ be a prime of $E$ above $\ell$, and let $\mathcal{E}$ be an irreducible lisse $E_\lambda$-sheaf on $X$, with corresponding representation $\rho$ of $\pi_1(X)$. Assume that, for every closed point $x$ of $X$, $\det(1-\operatorname{Frob}_xt,\mathcal{E}_{\bar{x}})$ has coefficients in $E$, and that $\mathcal{E}$ is de Rham at $\ell$. Folklore compatibility conjecture. For each rational prime $\ell'$ and each prime $\lambda'$ of $E$ above $\ell'$, there exists a lisse $\overline{E}_{\lambda'}$-sheaf on $X[\ell'^{-1}]$ compatible with $\mathcal{E}|_{X[\ell'^{-1}]}$. This is an existence conjecture for compatible systems over arithmetic schemes; the paper proves only some cases using results of Lafforgue and of Barnet-Lamb, Gee, Geraghty, and Taylor.