Rationality conjecture for Betti strata of normal functions

Let $S$ be a quasi-projective complex variety, let $f\colon X\to S$ be a smooth family of projective varieties of relative dimension $d$, and let $N$ be a relatively ample line bundle on $X$. Let $Z$ be a family of homologically trivial cycles of codimension $n$ in $X/S$, with $n\leq \frac{d+1}{2}$, which is homologically trivial and primitive on each geometric fiber $X_s$, and assume that $f$, $N$, and $Z$ are defined over a number field $k$. The associated normal function is denoted by $\nu_Z$, and $S^{\mathrm{Betti}}(t)$ denotes the Betti stratum of rank parameter $t\geq 0$. Rationality conjecture for Betti strata. The Betti stratum $S^{\mathrm{Betti}}(t)$ is defined over $k$ for each $t\geq 0$. This conjecture concerns the arithmeticity of loci defined by the Betti rank of a normal function; its status is unresolved in the supplied text.