Green–Griffiths algebraicity conjecture for zero loci of normal functions

Let $X\to S$ be a family of complex smooth projective varieties over a quasi-projective base, and let $Z\hookrightarrow X$ be a flat family of cycles of pure codimension $i$ that are homologically equivalent to zero in the fibers. Let $\nu$ be the associated normal function, a holomorphic section of the Jacobian fibration $J^i(X/S)$. Green–Griffiths algebraicity conjecture. The zero locus of the normal function $\nu$ is algebraic. This conjecture is the mixed-Hodge-theoretic analogue of the Cattani–Deligne–Kaplan theorem on the algebraicity of Hodge loci for variations of pure Hodge structures; its status is not resolved in the supplied source.