The relative Fontaine–Mazur conjecture for local systems

Let $F$ be a number field, let $X$ be a smooth variety over $F$, and let ${\mathbb L}$ be an irreducible $\overline{\mathbb{Q}}_p$-local system on $X$. Assume that there is a finite set $S$ of places of $F$ and a smooth model $\mathfrak X$ of $X$ over $\mathcal O_{F,S}$ such that ${\mathbb L}$ extends to a local system on $\mathfrak X$. Also assume that, for every place $v$ of $F$ above $p$, the restriction of ${\mathbb L}$ to $X_{F_v}$ is a de Rham local system. The relative Fontaine–Mazur conjecture. Then ${\mathbb L}$ comes from geometry: there exist a dense open subscheme $U\subset X$ and a morphism $\pi:Y\tilde{o} U$ such that ${\mathbb L}|_U$ is a subquotient of $R^i\pi_*\overline{\mathbb{Q}}_p(j)$ for some $i,j$. The conjecture predicts that good reduction away from finitely many places together with de Rham behaviour at places above $p$ forces a local system to arise from the cohomology of a smooth proper family, up to taking subquotients and Tate twists. It remains open in this general relative form.