Griffiths–Green singularity conjecture for normal functions

Let $X$ be a smooth projective variety, let $Hg^p(X):=Prim^{p,p}(X,\mathbb{Q})$, and let $L$ be a very ample line bundle on $X$. For a positive integer $d$, write $\bar{S}=|L^{\otimes d}|$ and let $\nu_\alpha$ denote the admissible normal function associated with $\alpha\in Hg^p(X)$.

Griffiths–Green conjecture. For every $\alpha\in Hg^p(X)$, the associated admissible normal function $\nu_\alpha$ is singular on $\bar{S}=|L^{\otimes d}|$ for some $d>0$. This conjecture proposed an inductive approach to the Hodge conjecture using singularities of admissible normal functions. Richard Thomas proved that, assuming the Hodge conjecture, every primitive Hodge class restricts non-trivially to a hypersurface; the conjecture itself is therefore resolved in the source's stated conditional sense.