Variational meta-conjecture for properties of 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 $X$. Let $P$ be a property of local systems conjecturally equivalent to being of geometric origin, and suppose that for some $s\in S(\overline{k})$, $\mathbb{V}|_{X_s}$ has property $P$. Variational meta-conjecture. For every $t\in S(\overline{k})$, $\mathbb{V}|_{X_t}$ has property $P$. The source presents this as a meta-conjecture and notes that important instances have been proved or studied, but gives no general resolution.