The motivated normal-function zero-locus conjecture

Let $\mathbb F\subset\mathbb C$ be finitely generated over $\overline{\mathbb Q}$, let $S/\mathbb F$ be a smooth quasi-projective variety of dimension $D$, and let $\mathcal H\to S$ be a variation of Hodge structure of weight $-1$ and level at most $2E-1$. Let $\nu$ be a normal function motivated over $\mathbb F$, and let $\mathcal Z(\nu)$ denote its zero locus. Motivated normal-function zero-locus conjecture. The zero locus $\mathcal Z(\nu)$ is at most a countable union of subvarieties of $S$ defined over, possibly different, finite extensions of $\mathbb F$. Under the same hypotheses, the strengthened statement asserts that $\mathcal Z(\nu)$ is an algebraic subvariety of $S$ defined over an algebraic extension of $\mathbb F$. These are presented as fundamental open questions concerning motivated normal functions and their relation to fields of definition.