Relative Fontaine–Mazur conjecture for semisimple lisse sheaves

Let $k$ be a finitely generated field, let $\overline{k}$ be a separable closure, let $\ell$ be different from the characteristic of $k$, let $\Lambda$ be a finite extension of $\mathbb{Q}_\ell$, and let $X$ be a smooth geometrically connected $k$-scheme. Relative Fontaine–Mazur conjecture. If $\mathbb{V}$ is a semisimple lisse $\Lambda$-sheaf on $X_{k'}$ for some finite extension $k'/k$, then $\mathbb{V}|_{X_{\overline{k}}}$ is of geometric origin. The conjecture is basically open, apart from cases such as curves over finite fields where it follows from the Langlands program for function fields.