Variational motivic conjecture for local systems

Let $\overline{k}$ be a separable algebraic closure of a finitely generated field $k$, let $f:X\to S$ be a smooth proper morphism of smooth $\overline{k}$-schemes, and let $\mathbb{V}$ be a semisimple local system on $X$. Suppose that for some $s\in S(\overline{k})$, the restriction $\mathbb{V}|_{X_s}$ is of geometric origin, meaning it occurs as a subquotient of the cohomology local system of a smooth projective family. Variational motivic conjecture. For every $t\in S(\overline{k})$, the restriction $\mathbb{V}|_{X_t}$ is of geometric origin. This is stated as a prediction of the relative Fontaine–Mazur conjecture and is described as completely out of reach.